This is my own interpretation of the ending, if you have anything thing else to say that I may have missed out or any interesting points please leave a comment thank you. Oh and massive spoilers as if the title didnt actually give that away.
Once the songbird brings down the siphon Elizabeths attitude seems to change instantly. They arrive in Rapture, another world, another door. Rapture is like Columbia, they are in some ways both the same like Elizabeth says the doors which lead to other worlds are “all different and all similar”.
Elizebeth explains that they all lead to the same place, so no matter what her and Booker do, they can never really change what is about to happen because it already has and already will. The walkway will also only allow them to walk down a certain pathway, Booker cannot choose another way, it is already set out in front of him and you must follow.
Once booker walks through another doorway he appears at his own baptism, after wounded knee, through this door Dewitt won’t accept the baptism, that it can’t wash away his sins and what he has done. Through the next door we walk into Bookers home, we know that he clearly turned into an alcoholic and more than likely a gambler from the amount of bottles which are on the tables.
Once Booker hands his baby over to Robert Lutece he replies with “Mr Comstock washes you of all your sins” showing that Comstock seems to think that he has the power to wipe someones sins when really this isn’t the case. The realisation that Anna is Elizabeth comes when they are trying to pull Anna through the tear in the wall and she loses her finger.
We appear at the baptism again, but this time things are different Elizabeth appears with Booker but she isn’t the same Elizabeth that he knew, her appearance is different, she hasn’t got a bird or cage pendent which you received nearer the start of the game. As each different Elizabeth appears there seems to also be one in an outfit that never appears in the game, which again tells us that there are more possibilities, more than we have seen.
Booker accepts that he is also Comstock, that in one life he took the baptism which supposedly made him a better man. But in reality he didn’t become a better person and his sins were never “washed away”, showing that the bad things which Booker had done could never be taken away and he never really changed. Booker and Comstock are opposite sides of the same coin.
Once the Elizebeths drown him in the baptism waters they begin to disappear, showing that none of this has happened yet. He hasn’t yet turned into Comstock but since this has just happened after the battle of broken knee there is still a chance that this could actually take place again.
There is a small scene after the credits of Booker walking into the room where Anna’s crib is supposed to be, “Anna is that you?” we never truly know if she’s there, I think that’s a question that is left unanswered. Has Booker already given her away and is it too late to change anything? Or has he realised his mistake and is Anna still there in time to stop what we have seen happen. This shows that everything can just happen again with infinate possibilities.