Considering that I can survive with only half of my brain, and I am a mind living in a body. If we would split my brain in two and then implant one part into some other body and leave other half in my head:
I'm not a neuro-scientist by any means, but if you are talking about straight up right lobe in one person and left lobe in another, it would get a little weird.
I would say philosophically, you would have two individual entities, which covered in another answer. Legally I think your stuff would get split up like in a divorce or something, if the two entities couldn't agree that is. I mean, you are essentially divorcing the two lobes of your brains. Either way that court case would get all up on national or even international news, and possibly into the Supreme Court.
Edit: I believe Dave already addressed some of the functionality of the two half brained entities with that video of Dr. Ramachandran, and his work. Which I will totally watch when I get home.
However for the straight up brain copy, you would still remain your individual entity. You would be you, and your stuff would be yours, and your job would still be yours (unless you work for a company that does not believe in cloning and by being a part of that experiment you would be going against the goals of that company). The new entity would also be it's own entity, as the previous answer stated, instantiated up to that point in your life. From then on it's free to make it's own choices, to do hard drugs, or become a mechanical engineer. The only thing I would say you morally owe your clone, or copy, is what anyone would owe any living thing they bring into this world, until it can provide for itself, or at least has the ability to. Which it already should, but a place to crash until it can get a job and find it's own place would be nice.
The follow up question in my mind becomes a moral one for cloning workers. Like you have this small workforce of people who spent their lives becoming masters of their craft, and now you can have as many masters of that craft you want, if you have the money of course.
I'll respond as if you'd ask what happens when you clone your mind into another substrate. I'm not sure what you mean by splitting your brain. Lobe-by-lobe, atom by atom, or by functional pattern? It's a vague question I'm not sure I understand.
But in terms of simply copying the state of your mind ... defined as, the state and position of every atom or quark of your body ... into another substrate, such as a computer, I do have a suggestion for how to think about that situation.
It would be like a process running in a modern operating system, creating a copy of itself with its own private copy of memory, initialized to the same state as its parent process. Initially you and the clone would feel the same; but over time, they'd have different experiences and be different people. They are in fact different people, they've just been initialized to the same state.
Same idea with Star Trek transporter technology. You are actually destroying the collection of atoms that make up Captain Kirk, and recreating the exact same patterns in a different collection of atoms at the target location. You are killing Captain Kirk and then creating a brand new person initialized to Kirk's state.
[Not his exact state, since the transporter had to measure his state, and the measurement would alter the state. Hey I think I just refuted teleportation!]
And having done that, you could of course clone as many copies as you like. None of them are the original Captain Kirk, whom you've just killed at the source. Or let him live. He can argue that he's the "real" one ... but they'd all feel exactly the same! Feeling like you are you, is just the state your atoms happen to be in. Change the state, change the feelings.
I posted this same idea in response to a similar question, here.
Assuming a mind could be uploaded, how would we know if it worked?
My response got zero points at that time. I'm tossing it out there again because I think it's the right answer. Or at least offers a potent metaphor for how mind transplantation works.
But I do think I've found a little flaw in the idea of a matter transporter. That deserves some more thought.
