Above everything else, for me, the original Bioshock had an outstanding atmosphere. The whole underwater metropolis concept was fantastic (still is today in my book). I haven't (re)played the game in a very long time though, and I know it might not be as good today as I remember it right now so I'll leave my memories where they were last created when I originally played the game. I normally don't like replaying old(er) games when I know that they probably didn't age well, only to have my embellished thoughts on them be tarnished by the harshness of new standards and the damaging of time. I loved Bioshock at the time, and it's all that matters to me.
The most recent one I played, and finished of course, was Infinite. I liked it, a lot (well overall, it's definitely not 'perfect'), and Elizabeth was (probably still is) the best 'sidekick' / following A.I. character I ever had the (pleasure?) to have at my side in a FPS game (or just in a video game in general). I remember being very hesitant playing the game when I learned via previews and some of the first reviews that we actually had an that A.I. partner following us nearly all the time (well, from a certain point on and then it's for the majority of the game). My first thoughts (before actually playing the game) about that 'follower girl' was things along the lines of "She's gonna get in my way all the damn time", or "She's gonna get stuck in some important scenes and I won't be able to progress anymore", or "Her voice is going to drive me crazy", or "She just won't STFU and I'll want to stop playing after 10 mins", or "I'll probably have to escort her in some inevitably-dumb timed mission where at the last moment a horde of enemies will overwhelm me and she's gonna die cause her A.I. will be equally stupid". Et cetera. I always expect the absolute worst from "followers" in most games, usually due to how generally-speaking A.I. is never clever and always gets stuck trying to walk up stairs or won't have the coding to jump above a stupid rock on their path, etc.
Anyway, long story short Infinite eventually went on sale on Steam and at that time I was actually looking for a 'good old' story-driven, mostly linear FPS game in the veins of Half-Life or indeed the original Bioshock and so on. So I put my hesitations aside and bought it. From the moment the game started I sort of mentality built up a shell, just to make sure that whenever I got to start playing with that girl around me I wouldn't just go bunkers immediately. I played... and played and kept wondering "So... when is that girl I keep seeing in the previews gonna show up...". Then eventually, it happens. And... and... wait. I remember thinking "Wait a second, am I actually... liking this?! Ok, this is getting interesting.". She had personality, good voice acting, and mystery surrounding her too? Hmmmm... ok let's continue on with this. And, as I kept playing... the most unexpected thing happened: I didn't even realize she was there for the longest time. That, in and of itself for me, was the proof that she had been VERY well tested internally. Her A.I. 'pathing' on all the maps and around the player is superb. She NEVER got on my way or blocked my view during action scenes. She never got stuck, she never got on my nerves. What sorcery is this?! You're an A.I. follower, you ARE supposed to be S.T.U.P.I.D. by video gaming LAWS. The ONLY, single other comparison I can make with Elizabeth that serves a very similar role to a main character in a video game is Alyx in Half-Life 2, and that's pretty much it, it just stops there. Anyway... yeah it sort of shocked me. She ended up being so well done that at some point I thought during my playthrough (but before I completed the game) that she'd probably be the main character of a potential sequel (or prequel) to Infinite. In the end, I found her more important and more interesting than Booker... and I don't think that was a bad thing per se anyway.
I mean, other than the fantastic art / visual style of Infinite (absolutely loved the 1912 'sky steampunk' feel and visuals) the one aspect of the game that obviously makes it stand out from the rest of the franchise (and from the rest of many, many other video games in general, but mostly other FPS games) is Elizabeth. But not just as a character for the story-telling portion of the game, but also on a purely technical side; She. Just. Works. It's well-tested, well-coded A.I. that does NOT do any dumb shit like 90% of all the gaming industry's A.I. does. If as a player you happen to be easily immersed in a game's setting / context / environment / story, etc (like me, usually)... then Elizabeth can definitely be not only very likeable (at least at some point, if not the case initially), but she's also acting in a believable way, and in accordance to her personality. I honestly don't think that she was a nuisance (very far from that, as I thought she would end up be), and for me to say that is quite the accomplishment (just speaking for myself, because if it wasn't clear up to this point I normally absolutely loathe followers especially in FPS games usually because they're very uninteresting to start with as characters but because their A.I. always makes a cockroach look extremely intelligent by comparison), so for that I have to applaud the developers. As as I am typing all this I keep getting some flashbacks of Infinite and how much I ended up liking her perhaps more than most of the rest of the game.
Actually, I do think that Infinite's story telling, story pacing and characters were very good. But, I have to say, that Infinite's actual game-play wasn't that engaging. And although I loved Infinite's visual style and presentation... I STILL definitely prefer the original Bioshock's underwater setting and atmosphere. All in all, I really enjoyed both B1 and Infinite pretty much equally I'd say, but for different reasons. As for Bioshock 2... I actually never played it, so I can't really judge that one. I remember reading some reviews, tried to skip as much media as possible at the time to avoid potential spoilers... but I never ended up buying it in the end, dunno, I guess it just escaped my mind. And from what I understand of the franchise Bioshock 2 was sort of a 'spin off', but I'm not sure about that (feel free to give me the info on that, I don't think I'll bother reading the game's Wiki page anyway).