How to solve the World 4 hunt?

Solution 1:

You have to kill every monster like in World 2. However, with time tied to movement, if you move to the left of where (and when) you killed a monster, you'll undo the kill, so you'll have to kill them by jumping on them and moving right.

So, move to the far left of the level and make a note of where all the monsters are. You'll have to kill them in order from left-most to right-most.

First, the one at the very top, then the one on the mid-left platform (by jumping using the monster on the mid-right, not using the ladder), then the left one on the bottom. Then the one on the lower platform on the right*, and then the one on the mid-right.

The door is time-independent, so it will stay unlocked from this point on.

*For the penultimate monster, you'll want to get a higher jump by jumping this one twice. First, jump on him moving to the left so you get the boost without killing him, and then finish the jump moving to the right so you end up on the platform while killing him.