Why would songbird capture our Elizabeth in the 2nd parallel universe when the Elizabeth of that universe was never saved by Booker (he died a martyr before he could reach her)? Was it simply because songbird happen to chance upon her? Also, shouldn't that universe now have two Elizabeths (this wasn't reflected in the game)?
Elizabeth’s capture (spoiler)
bioshock-infinite
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Well, there are multiple theories about this and only one that make somewhat sense:
The theories that she took it off or that it was a oversight by the developers don't make any sense. Even if the brooch doesn't change the plot it's still, weirdly, a important plot point.
The other theory is that by drowning Brook, Colombia doesn't exist and so stuff starts to... huh, disappear. Doesn't make sense either since Elizabeth would disappear sooner and the brooch was given by the Lucete. That might give it some "time wibbly wobbly" effects.
The last theory, that I know of, is that she wasn't Elizabeth. Not your Elizabeth. Booker confirms this in the dialog when he doesn't recognize her after some moments of confusion. This brings the question on where is Elizabeth or perhaps she is all of them.
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.
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My understanding is that Booker and Elizabeth don't actually travel between the dimensions. Elizabeth creates a superposition of two (or more) universes instead.
This is consistent with how tears work during the combat, how the Siren is brought to (un)life and explains why the guards in the club remember being dead.