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.