Elizabeth was Booker's daughter all along, and that's why his hand has the letters "AD" branded on it (Anna DeWitt, her birth name).

Now comes the really tricky part. Booker's voyage to Columbia in 1912 had nothing to do with his debts, because he'd already sold Elizabeth to the Luteces and Comstock back in 1893. He was just fabricating false memories using past events.

Why did he sell her off to Comstock in the first place? Well, now comes the REALLY tricky part. Comstock was actually Booker all along, but only the Booker of an alternate universe - where after the Wounded Knee massacre in 1890, Booker erased his past and his sins through a baptism, and was "born again" as Comstock (in the alternate universe shown in the ending, he refuses to be baptized and thus lives on as Booker, and then loses his daughter). Thus, he becomes a religious fanatic, a firm believer in American exceptionalism, and everything else you see embodied in Comstock. He never had a daughter, and he later became sterile. He builds Columbia, and turns it into what we see in-game.

Because he needs a successor who will "sit the throne" of Colombia after his death, Comstock (with the help of the Luteces), opens a gateway to an alternate universe and since he is sterile, "kidnaps" Booker's daughter, Anna - or as she will later be known, Elizabeth. Remember, he'd never had a daughter, and was rendered sterile by exposure to the Luteces' trans-dimensional equipment. Our Booker, the one who never accepted the baptism after Wounded Knee, is robbed of his own daughter by an alternate universe version of himself.

19 years pass, and Elizabeth grows up, Columbia is disavowed, the Luteces are betrayed and actually killed by Comstock, and Booker plummets into a downward spiral of self-destruction, and, perhaps, insanity. Again, for reasons I do not fully comprehend, the Luteces journey... to SOME universe, and offer Booker one last chance to redeem himself and save his destiny.

But as Elizabeth discovers once the Siphon is destroyed, things aren't quite that simple. This whole malarky has produced a massive inter-dimensional paradox, and the only way to do it is to... kill the original Comstock? Yes, it is - but who is the original Comstock? Booker, of course.

And so, Booker is drowned in the waters of the river where he had been reborn as Comstock. One after another, the different versions of Elizabeth vanish. In a post-credits vignette taking place in 1893, Booker appears to discover Anna, back in her baby crib - it would seem that inter-dimensional order has been restored: Elizabeth grows up with her real father, Columbia is never built, and everyone lives happily ever after.

Needless to say, it's an incredibly convoluted, poorly explained and nigh-incomprehensible ending to what should have been a much simpler tale. There are also many questions that remain unanswered... and that's what the Season Pass is for!!!