I think Booker is caught in some kind of time paradox. He meets a couple that asks him to flip a coin and it goes says heads which you then notice was done at least 200 times in a row. Plus when you meet them again they call themselves your friends after you try and shoot them. I don't know, just a guess.

The op is definitely onto something though. I specifically remember when you pick a side of the coin (say heads for arguments sake), they mention something along the lines of:

See, I told you he wouldn't pick tails, he never does.

Does this have something to do with Booker repeating this rescue time and time again within an infinite loop? Is there a third timeline where he is unable to save Elizabeth if he doesn't partake in the drowning of himself at the baptism. Is Old Elizabeth standing above a burning New York the result of a third timeline, wherein Booker is stuck in an infinite loop, unable to rescue Elizabeth causing her to age and take Comstock's place?