Colbert Bump: One trailer has given this to Nico Vega's song "Beast".
Dueling Games: Between this game and The Last of Us. Both games were hotly anticipated and feature a gruff, grizzled, guitar playing protagonist with a questionable past (both voiced by Troy Baker) who is tasked with escorting a young girl (with different variations of the same name: Elizabieth/Ellie) through dangerous territory and focus on the development of the relationship between the two. Upon their respective releases, both games were incredibly well received by critics and gamers alike, with the both of them being a stronger contender for the title of "Game Of The Year".
Fan Nickname: Skyoshock, Aeroshock, or Highoshock instead of BioShock Infinite, due to the game being set in a floating city.
Chronoshock for the timeline-jumping the game goes through in the second half.
"Disney Princess" for Elizabeth due to her resemblance to Belle.
What Could Have Been: Lots. Development had many, many, many people coming and going, so it was inevitable.
Originally the city was supposed to be darker and gloomier with a look inspired by Art Nouveau (think late 19th century Paris in the sky). This idea was scrapped both because it would make the city look too similar to Rapture and because it didn't fit with the later decided focus on American Exceptionalism. You can still find some Art Nouveau accents in the Emporia District.
The original conflict was also going to be between "tech geeks" and "luddites". Eventually Levine didn't think that these groups were "compelling polarities" (read: controversial enough) and thus went with the political angle.
According to this video, Levine heavily changed the character of Comstock after one developer, who was devoutly religious, threatened to quit his job over how offended he was by the character's portrayal. According to Levine, the entire experience inspired him to work the concept of forgiveness into Comstock's world view. He claims he didn't do it solely for the complaint, but because it would also make a better story.
The 2011 E3 gameplay segment features, among other things, normal American flags, Abraham Lincoln-themed toy (Lincoln is demonized in the final game's Columbia) and most notably posters of a clean-shaven Comstock who looks more like a cutthroat capitalist than a religious prophet. Booker and Elizabeth are trying to reach him to negotiate a safe passage from the city, when in the final game he is the biggest obstacle on their way.
Saltonstall, the bellowing man you see in the trailer, is almost completely cut out of the game. You hear some token references to him and find his scalp nailed to a board by the Vox Populi.
Elizabeth's supportive powers - such as calling lightning strikes down on enemies to stun them for Booker, were reduced to her opening tears in space and time, opening locks and finding supplies in battles. Also, having her use her powers would supposedly harm her, and the player would need to take this into account as it would affect how the story would play out, similar to harvesting and saving Little Sisters in the original BioShock games, and itself being an old mechanic of an unused BioShock conceptnote The original concept of plasmid usage was either Jack uses them more and more, becoming more monstrous and scarred, just like the Splicers, or refuse and keep his humanity, at the cost of safety.
Daisy Fitzroy was initially white - looking more like an expy of Sophia Lamb - and less nuanced - she was an anarchic intellectual, representative of hard-left revolutionary intellectuals in the 20th century.
Unlike promotional info, the Vox does not actually have any interest in Elizabeth at all. They're just interested in their fight with the Founders, with just the Founders interested in Elizabeth.
The Vigors were originally supposed to act as disposable and unreloadable powers of sorts, having a set number of charges depending on how powerful the Vigor was. This was supposed to force the player to keep thinking up new strategies on the fly, instead of spamming the same spell over and over again and forcing players to conserve their powers. The finished game uses the Salts system which is pretty much EVE only without the ability to recharge it at will.
The player would gain passive abilities by "Nostrums", which acted like Gene Tonics with the caveat that some were "unstable" and forced you to pick between three randomized abilities and others were "stable" and had no surprises but were more expensive. This was replaced by "Gear", which you can equip in four slots and is randomized once you find it.
Pretty much everything from the E3 2011 demo was removed. A few of the more memorable (and plot-relevant) things where just moved around though. The initial demo with Stephen Russell as Booker and Saltonstall had virtually everything removed except for the weapons and Murder of Crows vigor.
This is a reasonably common thing to happen with a game where Ken Levine was involved. If you go back and look at the earliest trailers for the original BioShock, a lot changed between the earliest builds and the final release, including the weapon designs.
In previews, the Silent Boys were played up as on of the Heavy Hitters you had to contend with, similar to the Motorized Patriots. However, in the actual game, they only show up in one of the chapters near the end, and work completely differently from how they were initially described. The Siren was also listed as a Heavy Hitter, but she's only a multi-part boss battle in the final product.
Word of God confirms that during development, the team had to cut "five or six games" worth of material from the final product.
The gameplay trailers imply that something is screwing with the citizens, who are completely apathetic to the fighting between the Vox and the Colombian military. In the game, you only encounter a few people afflicted with this, who are much more talkative than in the trailers, and they're a result of being killed in one dimension and getting fused to a still-living version of them in another.
Gameplay also implied some people were dimensional hoppers, as Saltonstall briefly becomes a communist and heavily anachronistic pop culture references pop up from the Vox (one tells Booker to "keep moving, Kemosabe"). In the final product, people do dimensionally hop, but unwillingly, and it usually leaves them insane.
Speaking of the citizens, many of the more innocent ones are being harassed by various Vox Populi members. Booker can choose to help them out at the cost of angering the enemies (and the rescuees would repay him in some way). Only at the raffle at the beginning of the game does this ever happen, and it's a quicktime event. However, some content of that demo does reappear in the final game, albeit modified (the postman execution cannot be prevented).
Thanks to the Art of Bioshock Infinite art book, we get to see even more cut content, including but not limited to:
"Merged" Splicer-esque civilians horribly distorted thanks to tears.
"Vigor junkies,"◊ also Splicer-like enemies (although the designs for what appears to be a Shock-Jockey-based junkie were actually used as Frosty Splicers in the Burial at Sea DLC)
Stand-ins for the Songbird, including toy-based automatons and a flying Big-Daddy-like creature nicknamed the "Mothman."