Like when his hand gets chopped off and then he magically grows a new hand right away. Then a little while later he says: "My new hand, it's a fightin' hand!" and he sounds like a totally uneducated hick from a trailer park in the USA as he is saying that. If that's not schizophrenic then I don't know what is.
That was supposed to be a joke. Might be British humour.

As for what I consider Doctor-like: to the best of my recollection the Doctor is supposed to dress and act like a professional and intellectual, and have a decent amount of hair.
Ok then. Personally I think of the Doctor as an eccentric genius.