The best adventure games like BioShock are BioShock Infinite, Prey and System Shock 2. Below are 7, each with the reason it is here — and, at the end, what filtering to adventure costs you.
Filtering to adventure means the place and the story are what you want back, not the challenge. It is the right filter when the thing you remember is where you were rather than what you did.
What you get: A world with a reason to wander it, and writing that survives being read rather than skipped.
What made BioShock worth playing in the first place is covered on the full list. This page is the adventure part of that answer, and only the adventure part.
The sequel in spirit, and Columbia is as strong a setting as Rapture.
The best immersive sim of the last decade, and the closest thing to a BioShock successor.
Where BioShock came from, and still frightening.
Powers plus a city with a coherent, dreadful history.
For the underwater setting and a story that genuinely unsettles.
Environmental storytelling as the whole design philosophy.
A building that is a character, and powers to tear it apart with.
Mechanical depth. Several of the best adventure games have combat that is deliberately unsatisfying, because they want your attention somewhere else.
Specifically, these are strong answers for BioShock that are not adventure games, so they are not on this page:
The full list is on games like BioShock — 9 picks, no genre filter.
BioShock Infinite. The sequel in spirit, and Columbia is as strong a setting as Rapture.
No. BioShock released on the Xbox 360, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Anywhere the story and the place matter more than the skill ceiling — point-and-click, walking sims, narrative thrillers and most open worlds. The label alone tells you very little, which is why each pick here says what it actually is.
They split hard. Open-world adventures run to a hundred hours; the narrative ones are often finishable in an evening, and the short ones are usually the better written.
By hand. Each pick was written for the full BioShock list first, on what the game actually does, and this page is the part of that list that is adventure — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the adventure guide for where to start in the genre generally.