I think I could definitely say that if one's personality were completely changed, then one would cease to function as the same identity and would instead be someone new, even in the same body. And -- to head you off before you ask -- yes, I believe the reverse is also true: the same personality (that is, the same mind) transferred over to a new body would retain the same identity that had previously occupied the original body.
BOB: Now you've complicated things -- is identity of the personality or the mind? Or is the mind the seat of the personality, and also identity? In our first supposition of one who suffers a trauma and undergoes a personality change, suppose also that the memory is unaffected. Would identity have changed here, even though the two personalities share a consistent history?
CIN: Yes, I think that would be a fair assessment -- if one perceives the world differently, and interacts with the world differently (and after all, what is personality but perception and interaction?) than they used to, then it would be reasonable to say that one had changed identities.
BOB: Then if you were (fortune forbid) to suffer a personality-changing trauma that kept your memory completely intact, you would not consider yourself you?
BOB: So you would have memories of having been someone else, but would not actually be the same person -- thus two identities would have occupied your body at distinct times? I see you hesitate, which makes me glad, for though I do not wish to think that our identity is tied solely to our bodies, there is obviously more to this question than simply what lies beneath the surface. Let's see; so far we have established that identity is not fully dependent on the physical body, and yet is in some ways attached to it -- namely, we have discovered that the personality is detachable from the body, but the inverse of this is not true. So far, however, our discussion ahs been limited to ideas of discrete identities existing within discrete bodies, but what of multiple simultaneous identities in a single body, or a single identity in multiple bodies?
CIN: Would it surprise you to hear that I don't quite follow you?
BOB: I don't even know where I'm going; it certainly wouldn't be wise to follow me. But at least play along -- suppose your mind was somehow projected into two different bodies, neither one of which was the one you now occupy. Would either of those two bodies be you? Or would they be new identities? Or would they both be you? Could they both be you?
CIN: ....well, aren't you going to answer?
BOB: I'm not really sure myself. But because we have determined that personality can be removed from he body, though not the other way around, it seems reasonable to assume that at least one of the new bodies could be an extension of your own identity.
CIN: And the other body?
BOB: For now, I would say that at the moment of conception, as it were -- that is, at the moment when your mind/personality was transferred to these two new bodies -- at this very precise moment than yes, the two identities were both you. And from this moment forward, neither identity was you, but rather a distinct personality with a shared set of memories to a certain point, and an independent history thereafter.
CIN: But before, when I said that memory wasn't really a part of identity, you said
BOB: Yes, I know what I said. Unfortunately, I'm on my way to a movie and can't really stop to argue with you any longer. Besides, greater minds than ours have tackled this same problem (in far more complex ways), and the issue still isn't resolved. I just taking comfort in knowing that I'm me right now -- though I suppose I might have a transformative experience in the cinema.
CIN: If you don't suffer an unexpected head trauma first...