I disagree that I disappear during states of "unconsciousness". Sleep/general anesthesia/drugs are just different states of brain activity, but I'm still "me" during these states. We don't say people become no one, or someone else when they sleep.

So, a copy of your consciousness (if such a thing is possible) is different than sleeping. No matter how much the clone believes it is me, it will never be me. I still die in my scenario, even though the other feels differently about it.

To add to that, asking "who am I" is a philosophical question, and I'm not a strict empiricist, so I don't categorically dismiss the idea that there exists an "I" outside of my natural body.