Stephen Levine wrote an excellent book titled, "Who Dies?" (Ref. 1) which explores the question, "Who am I?" He asks the reader, for instance, to imagine loosing body parts and functions one by one and try to identify when you cease to be "you." Perhaps only your brain is left -- are you still "you?" If the brain is then pared away, as it is with Alzheimer's, are you still "you?" Is it your memories that make "you?"
References:
1. "Who Dies" by Stephen Levine