So if an alien abducted you, created a clone to replace you while you are experimented upon, you'd still agree that you are living normally, and nothing happened to "you"?
Few years ago, I was in a PR show for a shopping center where Paris Hilton showed up. Everyone was interested to see her. It turns out she was not the true Paris Hilton but a doppelganger.
Conversely, people usually go through profound behavioral transformations several times in their lives. Does that mean they are not "themselves" anymore?
Very broadly speaking, I would say yes. Again this is just my take, but I argue that different sets of behaviours are indicative of different identities.
For example, at work I may keep to myself and be very quiet - in that setting my actions would define me as an introvert. With my friends I may be more outgoing and talkative, which is more characteristic of an extrovert. So I have two identities depending on my behaviour. I am not my introverted self when with friends, and not my extroverted self when at work.
Interesting! I would say that I am living both normally and abnormally since "I", in this case, is referring to two entities which are indistinguishable in terms of their behaviour.