Why do I specifically cite the E3 2011 version of Bioshock Infinite? Because a lot of what we see here could easily have worked as a multiplayer mode. Calling up tears that have to go on a recharge timer? Check. Using Vigors more like special grenades than as Plasmids? Check. Massive maps with numerous vantage points, skylines, and destructible pieces (such as the blimp)? Check, check, check.
Bioshock 2: Fall of Rapture had an outstanding multiplayer mode, which is why people still actively play it even though Bioshock 2 released in 2010. The fact the best Irrational Games could come up with for Infinite was a pinball-style tower defense mini-game and a co-op mode (both of which never made it in the final game) is depressing. Granted, I realize my suggested alternative isn't the farthest out of the box.
Really, the Fall of Rapture framework, other than specific mechanics like hacking turrets/vending machines, is just fine as a template here. Narrative provided through the player's apartment and audio diaries works as well in Columbia as it did in Rapture. Unlocking new weapons, vigors, gear, and weapon modifiers fits just fine as a progression system. Even maps with little winks and their own narrative touches is part of what made Fall of Rapture feel like a true Bioshock game, so include similar level design here.
Really, all you are doing is trading mechanics. Instead of hacking, include Skyhooks and skylines to ride. Instead of a Big Daddy suit, either have a random Heavy Hitter that can be triggered, or a one team can use the combat blimp like the Gunship in Call of Duty. Make the levels a bit less cramped and... ladies and gentlemen, I present to you, Bioshock Infinite's multiplayer.
Much like with Fallout 4, it's just not that complicated to see a multiplayer mode working. I'm sure something more imaginative could be accomplished in a cooperative space, but that would be limited either in replayability or in the amount of content easily made. For Bioshock, competitive multiplayer works surprisingly well, so I'd say stick with what works.
Is there any game you wish would add multiplayer? Have an alternative idea for the ones mentioned above? Let us know in the comments below!