Timeline A: After Wounded knee, Booker washes away his sins and is born again. Turning to God, Booker creates Columbia with the help of the Lutece's, due to his spiritual transformation he changes his name to Comstock and successfully invents time travel (also with the help of the Lutece's). Time travel works but such close proximity to the testing of it renders Comstock (formally Booker) sterile.
Unable to produce, and realising he'll need an heir to his throne if Columbia is to continue after his passing, he goes back in time with Robert Lutece and helps an alternate timeline Booker (Timeline B) to clear his debt in exchange for Booker's daughter. Booker reluctantly agrees and hands Anna (who is renamed Elizabeth) over.
To protect his secret and the real identity of his newly-purchased daughter Elizabeth, and to hide the fact that Elizabeth is about 1 yrs old at this point AND that Mrs. Comstock in the nine months prior showed no signs of pregnancy, Comstock makes up an elaborate story that Elizabeth was the product of a miracle and was conceived in just 7 days. Not only does this hoax work, but adds even more credence to the religious dream he's peddling with Columbia.
Meanwhile, enraged that the child isn't hers, Mrs. Comstock is teetering on the edge and believes Comstock has had an affair as she's damn sure the child hasn't come from her (as heard in the audio logs). Slowly becoming a risk and worried that she'll compromise things and blow the facade to pieces, Comstock has Mrs. Comstock killed and frames Daisy Fitzroy for the murder, which again, only helps Columbia's racial stance and fuels its "pure" beliefs.
Paranoid, Comstock also has both Lutece's killed, erasing all coconspirators in his big fat secret.
He then has Elizabeth locked away in a tower, probably due to her tendency of opening up tears. Comstock builds the tower and the siphon to stop her from doing this. (Thanks @khazidhea!)
Weary that Timeline B booker will return one day (I guess this is one of the reasons she's locked up and guarded by Songbird), he skews his religious propaganda that one day a False Shepard marked with the initials "AD" will try and lead his lamb (Elizabeth) astray. So that by the time Booker turns up enough of Columbia will be brainwashed and he will have essential raised an army to "deal with" Booker – thus keeping his secret safe, and Elizabeth none the wiser.
Timeline B:
After Wounded Knee Booker was almost born again but refused to go through with the ceremony as he didn't believe his sins could simply be washed away. After this he has a daughter (Anna DeWitt), turns to gambling and gets in debt with the wrong people. Robert Lutece shows up one day on behalf of Comstock and offers to clear the debt in exchange for his newborn. Booker, realising what a horrible thing he's just done, tries to get Anna back but fails, causing Anna to lose part of her pinky finger.
For the next say 19 years Booker goes through depression, and through his stupor engraves the initials AD (Anna Dewitt) onto his hand to remind/punish himself for what's he's done. The Lutece's both travel back in time after the exchange of Anna/Elizabeth (but before their deaths I believe) and guide a now damaged Booker to the lighthouse to "Bring us the girl and wipe away the debt". I'm not sure why they do this, some help here would be appreciated.
Booker goes to Columbia, attempts to save his daughter, but is unable to leave Columbia. I think this is due to being stuck in a loop, and the only way to save her from going through 19 years in Columbia is to drown Booker to prevent Anna from ever being born. Drowning Booker at the source also prevents Comstock being born out of the ceremony and the gambling debt from incurring in the first place.
Booker is killed in the river and therefore Anna/Elizabeth is never born.
