Here's a thought experiment I haven't heard before. I'm not sure if it's novel, but even if it isn't, it's definitely less well known than the popular thought experiments about cloning and the nature consciousness.
The question of whether it is possible to preserve a person's mind and consciousness is a well known idea in speculative fiction and transhumanist thought, as well as touching on more general philosophy and the hard problem of consciousness.
The first thought experiment to consider is one many people have already played around with on their own. What happens when you create a perfect copy of a brain? If you copy a brain, molecule by molecule, atom by atom, even electrical charge by electrical charge to achieve the same state, will the object you create be conscious? Would it have the same memories and ideas as the source brain?
Assuming no hidden metaphysics that somehow breathe consciousness into our human beings, the intuitive answer to the above questions seems to be yes. If our intelligence, memories, etc., are all just based on brain structures and electrical charges within the brain, a perfect copy should function exactly the same as the original.
Now suppose we clone your entire body, brain and guts and all. Is this clone also "you"? I think the intuitive answer here is that no, it isn't. You would end up with two different seats of consciousness. Each with exactly the same memories and ideas, but still different individuals, like identical twins on steroids. You wouldn't be able to see through your clone's eyes, hear their thoughts. If you were killed, it's not like your consciousness would be preserved in your clone's body.
This so far has been fairly common thought experiments that probably most fans of sci-fi had encountered at one point or another. This is for example related to a common depiction of teleportation, in which the teleportee's body is perfectly scanned and destroyed on one end, then rebuilt molecule by molecule as an identical copy on the other end. Personally I would never use such a device — while I might appear as the same "he" to my friends after teleportation, I don't see any reason to believe the person on the other end would be the same "I", and I myself would be dead.
Now comes the "linked brain" part. What if the clone, instead of being built as a separate person standing next to you, was initially built as an extension of you? What happens if we first build a second brain attached to your own? Let's say the two brains are connected at the brain stem, allowing for communication between them. We create a siamese twin for you, joined at your brain stem and we give some time for your consciousness to "spread" to both brains.
Eventually, we separate the two copies. Which one is now "you"? Who is the other person and at which point did they become a second consciousness?
Maybe split brain studies can shed some intuitions on these questions. In certain experiments, split brain patients appear to be inhabited by two separate conscious entities where there once was one. Maybe there's no such thing as consciousness and "conscious" thought is just a weird sensory artefact.
I for one definitely feel like I have a singular (though multi-faceted) "I" inside of me. But maybe I'm just not enlightened.
I guess this thought experiment could also be a modified to an "upload your brain to a computer" variant. Suppose we can perfectly simulate a brain inside a supercomputer. Let's also assume we have a perfect brain-computer interface (the 1000th generation of Neuralink) and we link a living human brain to a simulated replica of a brain on a super computer. Can these merge into a single consciousness? What happens when you unplug?