The game that we are playing right now is nothing what we were meant to play, but merely a thing that was chopped up and changed around scarcely 6 months before the release
There was two major revisions, in the earliest demos the game was going to be Elizabeth and Songbird. You'd get your powers using Elizabeth, which would harm her, (in later videos there were vigors too, and you could combo them sort of). Elizabeth was trying to escape Songbird, but had "Stockholm syndrome"-like relationship with him, so fighting him would ruin your relationship (this is sort of in the game still, Booker wonders if Elizabeth will let you take him down in the loading screens).
Elizabeth was nerfed compared to what she was supposed to be (having the ability to fight and the ending changed depending how you used her in combat)
She would also toss enemies into tears when prompted. If you used her powers to much her nose would bleed and she would get dizzy. (and die if you weren't careful)
Ask Elizabeth to kill also changed her demeanor as game progressed (if Elizabeth only kills and you don't she would call you weak and a coward... over time she wold kill on her own with your promoting signifying that she started to enjoy killing also effecting the ending)
In later ones (the 2011 footage, the one with the Lincoln hat) Elizabeth manipulates tears instead, but has trouble controlling them (she still does in the final game). The biggest difference is Comstock and the city. Comstock was just a politician (Liz and Booker were actually trying to reach him for help), and the city was more like Vox world to begin with. Also, the city was bigger and more open, along with enemies using the skylines to attack, not just when you're on them.
On Elizabeth herself -she was not meant to be Booker's daughter - her finger is there in all vids up until the most recent screenshots of her in different clothing and with Comstock being a third person who wasn't related either to Booker or her.
The name "Anna" may be a late reference added to the game that refers to the cosplayer who was noticed by the studio before release of the game for her stunning resemblance to Elizabeth, Anna "Ormeli" Moleva., which was also contacted to work with Irrational for promos around August 2012 - again the same time when Levine started doing the rewrites.
It was also in August 2012 when mo-cap work for Elizabeth begun and all previous work, voice acting was either cut or fused to make it to the deadline what was repeatedly pushed away further and further.
There is also a lot of stuff on multiplayer component that seemed to preoccupy people at Irrational for quite some time, and turned out to be a doomed piece of work before people started dropping away and that was yet again, in August 2012. 2K started pressuring Levine for results at around latter half of 2012 (what a coincidence!) and the combination of pressure and team loss might have prompted wholesale game change.
Julian Murdoch, who is friends with Levine and often runs podcasts which include him, tweeted that (@GWJ Rabbit) "Wow, you weren't kidding when you said you'd rebuilt the story from ground up in the last 6 months! Holy Cats!"
Sourceshttp://www.pcgamer.com/2013/01/22/bioshock-infinite-character-changed-religion/http://news.softpedia.com/news/Ken-Levine-Explains-Comstock-Change-for-BioShock-Infinite-333690.shtmlhttp://kotaku.com/5933119/bioshock-infinite-update-multiplayer-modes-cut-gears-maestro-joining-should-fans-worry ----->
http://pastebin.com/M2Qsiu1Bhttp://www.lazygamer.net/xbox-360/is-bioshock-infinite-in-development-hell/http://www.youtube.com/watch?v=IaqACTgXpJM - Ken Levine talks about early game
Game demos (2010 and 2011):
http://www.youtube.com/watch?v=j8z6FOL0G1chttp://www.youtube.com/watch?v=bs41eOXhwPA