Do we die when we jump off the edge?

To answer your question in the title, without spoiling the game's amazing storyline, yes and no.

This little "feature" actually ties in very well with the entire plot of the game. So press on and it will all make sense! And it's worth finding out, believe me.

The below section contains spoilers! Proceed with caution!

Ok, so to explain in more detail: The game is all about alternative realities powered by variables and constants. That is, because of the invention of the Lutece's device to travel among parallel universes and Elizabeth's power to open Tears, there are in fact, multiple Bookers in different dimensions of space and time. Each world sharing certain things (constants) and each world being different based off the choices Booker makes (variables).

In this case, jumping off Columbia could be considered a variable. Somewhere in another reality, a Booker ended it all and died off Columbia. But, because of Elizabeth's tears (the zapping sound), you "remember" your death in another reality. That's why your character feels hazy and confused just after you jump off. "What is this?" You are remembering your death, but you're not dead??

Just to clarify that it is not in fact Elizabeth opening a tear while you're jumping, to save you. (Since you can still jump off the edge when you are not with Elizabeth and this will still happen.) Instead, you can think of it like this: In that moment right before you jump two universes are created: the one where you jump and the one where you didn't. The Booker who jumped is dead, but there is a Booker who did everything you just did up to the point of jumping, stood still for a moment and decided not to jump, and that is the Booker you now are playing.

This explains parts of the game where you travel through tears, and combine two worlds into one. It gets really messed up then, because if in one world, a person is dead, and in another world a person lives, they both combine and result in a living person with a bloody nose who remembers that they died but are still alive. "How on earth would you reconcile that?"

And as far as I know, there is no advantage to jumping off Columbia besides witnessing glitches