It sounds like a dopey question. I'm me because I'm the same person I was yesterday, more or less. I have the same memories (except for the memories of the things that happened today), and I look more or less the same.
What if an exact -- molecule by molecule -- copy were made of me? A copy so exact that neither the original nor the copy could tell -- could have any possible way of telling -- which was the real me, and which was the copy? Would he be me? Would I be him? Or would we be two different people? Sure, we would look like two different people, but we would have the same memories and we would look identical.
Wikipedia: Identity of indiscernibles, the ontological principle that states that two or more objects or entities are identical (are one and the same entity), if they have all their properties in common.
Multiverse Quantum State Collapse