How is the after credits scene possible? (Spoiler)

How is it possible that Booker apparently remembers the whole Columbia experience?

In the scene he was quite frantic as to whether Anna was in the other room. How is this possible given that about-to be-baptised-booker drowned himself, therefore removing any possible strain of Colombia or Comstock to exist?

Solution 1:

Temporal paradox (also known as time paradox and time travel paradox) is a theoretical paradoxical situation that happens because of time travel. A time traveler goes to the past, and does something that would prevent him from time travel in the first place. If he does not go back in time, he does not do anything that would prevent his traveling to the past, so time travel would be possible for him. However, if he goes back in time and does something that would prevent the time travel, he will not go back in time. Thus each possibility seems to imply its own negation - a type of logical paradox.

- Taken from the Wikipedia page for Temporal Paradox

...I've come up with a theory that could explain the ending a little better: within the Bioshock world/universe/whatever you want to call it, paradoxes are impossible. This is explained, albeit somewhat vaguely, in one Rosalind Lutece's voxophones:

Our contraption shows us the girl is the flame that shall ignite the world. My brother says we must undo what we have done. But time is more an ocean than a river. Why try to bring in a tide that will only again go out?

Time is more an ocean than a river.

That is easily one of the most important statements in the game. What it means is that time is not linear; everything happens all at once. That is the meaning behind "lives, lived, will live" and "dies, died, will die". Everything that is happening, has happened, and will ever happen, is happening right now, but in a different universe - an infinite amount of universes, for that matter.

So, what does this have to do with paradoxes?

Paradoxes can only occur when time is a "river", meaning linear. Here's what the major events of Bioshock Infinite (and the ones that preceded it) would look like if time were linear As I wrote in the image, a paradox is created when Elizabeth drowns Booker. Booker dies, Anna is never born, Elizabeth doesn't kill Booker. Booker lives, Anna is born, Elizabeth kills Booker. This goes on for an infinite number of times.

But you already knew that, right?

The thing is, as Lutece said, time is not a "river', it is an "ocean". Here is what the events of Bioshock Infinite look like as an "ocean". What this means is that everything that ever happened and will happen happens in a separate universe, totally independent from anything else that has ever happened or will happen. One event does not affect another - at least, not in such a way that it could create a paradox.

Clearly Elizabeth can affect other universes, but a paradox simply cannot be created because in order for a paradox to exist, time must be linear. This is because when time is linear, events that happen in the past affect and determine what happens in the future. When time is an "ocean", there is no past, present, or future, so there is nothing to affect. I know, it's confusing, but it does make a sort of sense.

Now, how does this affect the ending and the post-credits scene? Because the ending scene of Booker's death has absolutely nothing to do with time. Well, that's not entirely true. The only importance time has on the final scene is that Elizabeth had to take Booker back to the time before he finalized his decision on the matter of baptism. Aside from that, time is irrelevant. We know this because the Booker that Elizabeth drowns is not 16-year-old Booker; it is the Booker that we have been playing as for the entire game. It could have been any Booker, really. The only thing that matters is that a choice is never made. Why? Because universes are created from choices. We all know that Booker became Comstock because in one universe he chose to accept the baptism. So, to eliminate Comstock, Elizabeth drowns Booker before the choice is made. He does not choose to reject the baptism, and he does not choose to accept it. When he is killed, he is still in the process of deciding. No choice is made, no new universes are created.

Booker's death does not create a paradox because it does not affect anything else. His death is independent from every other Booker in every other universe; the only thing that has changed is that a decision is never made. His death is a variable, whereas his lack of decision (sorry for the incredibly choppy wording there) becomes a constant.

...Booker's death is not a variable, since that would mean in another universe he does not die. Comstock's death - or technically, his lack of birth - is a constant because he is eliminated. Once all possibilites of Comstock's "birth" are gone, that's it. It doesn't matter if Booker continues to live, because it already happened ("dies, died, will die). Booker's death, on the other hand is a variable in the sense that he gets to live because no paradox was created, even though he technically died. If that makes sense.

[Booker] was drowned before he made his decision. Like I said, the only thing time has to do with it is that Elizabeth took Booker into the time when he is still deciding wether or not to go through with the baptism. Elizabeth kills Booker before he makes a final decision. Since Booker did not decide to go through with the baptism or run away, there are no new universes that are created. This is because universes are created through variables, and Booker's choice was a variable. Elizabeth eliminates the variable by killing him before either choice is made. This is proved at about 15:45 in this video. Elizabeth says, "Smother, before the choice is made. Before you are reborn."

Elizabeth kills Booker, but that is not the constant. She only kills Booker because she has to in order to do what her actual goal was: kill his choice. One Booker in one universe is dead, but all Bookers across all universes can no longer make the choice to become Comstock or remain Booker. If Booker cannot choose to become Comstock, then the only option is to stay in his current state (i.e. as Booker). Think about it like this: you are wearing a black shirt. Someone offers you the chance to wear a blue shirt, but before you decided which shirt to wear the choice is taken away from you. You can no longer choose to wear the blue shirt or continue wearing the black shirt. The only possible path that can occur from this point is that you continue wearing the black shirt not because you chose to, but because you were forced to.

^{Source: "Theory: Paradoxes do not exist" post at the Reddit /r/BioShock subreddit by Reddit user, lolmaster2000}

Solution 2:

I think this is actually the most confusing and difficult part of Bioshock Infinite in that the post credits sequence doesn't really make a huge amount of sense given what has come before. The last words before the drowning are:

An Elizabeth: He's Zachary Comstock

Another Elizabeth: He's Booker DeWitt

Booker: No, I'm both.

We can be reasonably certain therefore that Booker's drowning wipes out all of the Comstock and DeWitt timelines, because it happens before the fatal choice. If Booker DeWitt ever refuses salvation, then Zachary Comstock must accept it. One cannot exist without the other, which is why Booker has to be bought by Elizabeth to a point where he will accept his own death. The Elizabeths must push him under the water and drown him, thus destroying their own existences.

So, who or what then is the Booker DeWitt in the post credits sequence? There is of course a paradox inherent in the fact that if the Elizabeth's didn't exist because they drowned Booker DeWitt, they cannot have drowned Booker DeWitt. His existence after the drowning is in fact a direct negation of what came before the credits - his very need to die to save the multiverse from the monster that is Comstock. Perhaps it is this very paradox that allows him to live, although that doesn't explain why he is returned to the time when Anna was a baby.

I can think of three other possible explanations

1. The entire Bioshock Infinite experience is simply DeWitt's guilty conscience regarding Wounded Knee creating a fantasy of an alternate Booker. Effectively the "it was all a dream" ending. While logically consistent in that in this version of events nothing happened outside of Booker's dream, this would be a deeply unsatisfying ending, and robs the rest of the narrative of much of its power. It also doesn't really explain the presence of Rapture. I'm therefore going to suggest that it is not what was intended.

2. This Booker is from a timeline where he never went to the baptism ceremony at all. His life proceeded among very similar lines to original Booker's because, like original Booker, he never accepted forgiveness. It is an indication that in some sense, a Booker survived and stayed with Anna as she grew up. The thing is that such a Booker would have no continuity with the in game Booker, and it doesn't really make sense to show us this stranger.

3. My preferred explanation is that therefore Booker's Elizabeth and/or the Lutece twins were able to extract the original Booker from the timeline as he died, and give him an existence either outside time and space as theirs were, or shunt him sideways into a reality where he was not able to make his big mistake because Comstock no longer existed. So this Booker both died and didn't, just like the Lutece twins. He remembers all that happened to him, but is given a second chance as a reward for preventing Comstock. Again, this isn't fully satisfactory in that it takes some of the narrative power away from the ending, but I think it does work within the metaphysics of Bioshock Infinite, and furthermore explains why Booker appears confused when he is returned to reality, as he was doubtless expecting it to end.

This answer is necessarily speculative, but hopefully clarifies things a little.

Solution 3:

Needless to say — SPOILERS!

But the Booker we played was the one that was drowned...right?

"The mind of the subject will desperately struggle to create memories where none exist..."
– R. Lutece

So, throughout the game, we see instances of Booker's memories getting rewritten, as well as some other people. Whenever this happens, it is indicated by dizziness and bleeding from the nose. It's hard to draw a conclusion when only Booker, Chen Lin, and the guards having been affected, but I'll do my best.

Whenever Elizabeth opens a tear, she is moving to a universe that contains the conditions that she wants or needs at the moment. However, her ability also has a small area of effect, bringing with her people around her in a small radius. We can see this shown when Fink's Head of Security is unaffected, despite being dead in the first timeline. Whenever a tear is opened and the people step through, the mind will attempt to settle paradoxes by rewriting memories.

The first paradox we see is the state of being alive or dead, which affects Chen Lin, Booker, Lady Comstock, and two guards. What happens here is that the memories of the dead and the living are in conflict with each other, so the mind will attempt to combine memories of death with life. The effects of this seem to vary in intensity and results. A common ailment is that people will end up feeling traumatized and deeply disoriented, often seen struggling with their deaths. A possibly more intense reaction is the formation of new abilities, although Lady Comstock's situation is a bit special as 1) Elizabeth's powers were forced by the Siphon and 2) the Lady Comstock brought over is the one that encompasses the traits that Elizabeth imagined her to have.

Booker's situation is a special case as he is actually stepping into another universe. Thus the memories that merge will collect in his mind, which is seen in the case where Booker has memories of leading the Vox revolution. What is interesting is how Booker is unaffected when he's jumping from the first to second timeline in The Good Time Club. My guess is that because Booker is stepping into a timeline where there isn't a Booker. However, because the circumstances would've been the same, the mind does not have to rewrite his memories. This idea is tenuous at best, and could simply be the result of an oversight by the developers unless someone can explain this better.

The second paradox is the state of existing or not existing in the timeline, which is what affects Booker and Robert Lutece. Because Booker steps into a timeline where he doesn't exist, Booker's mind is trying its best to rectify that, muddling his memories to fit within this new universe. Robert Lutece also experiences this as well when he crosses over to meet his sister. This is a type of paradox where the memories of the same people will not merge.

Why is it the second paradox people will merge, but not the first type? Well, I can theorize that it has to do with whether or not the memories merging belongs to the same people. The Chen Lin in the first and second universes have the same identity and are practically similar, so their memories will merge. However, Booker, Comstock, Rosalind, and Robert are largely unaffected by the paradox of being in two places at once, with some minor hemorrhaging on the side of Booker and Robert. This may be due to the fact that there is enough of a difference between them that the universe allows them to exist simultaneously. In the case of the Lutece twins, what separates them is their gender, where combining them would probably induce another paradox of being both male and female. For Booker and Comstock, it is mentioned that Comstock's own identity has been eroded due to repeated exposure to the tears, thus making his identity different to that of Booker's.

So, what about Elizabeth? Why is not affected in any way? Again, we have a special case where Elizabeth's power over the tears allow her to retain her identity and memory. We cannot say for sure, as the origin of Elizabeth's powers is simply explained with a hand wave of being in two places at once, and the game doesn't delve deeply into this.

So, what does all of this have to do with the ending? Well, at this point, Elizabeth has the ability to open any tear she wants and can access the infinite doors that lead to different timelines. So, when Booker steps through, he is essentially going back and time, his own self is reverted to that time period's Booker. This can be seen when you accept Preacher Witting's hand for baptism, Booker's own hand is young and has no AD or bandage wrapping around it. This signifies that Booker has merged with his younger self, starting again at the baptism scene. This points to the fact that the Booker that we play through and the Booker that is drowned by the Elizabeths are one and the same person.

^{Source: Reddit /r/BioShock post by Reddit user awchern
This answer was partly copied (shamelessly) from another answer of mine here.}