I'd even go so far as to say that the me from 10 minutes ago isn't the same me as now. Consciousness isn't continuous, it arises and ceases. It's our memory that gives us the illusion of continuity.
And this is what makes me absolutely unconcerned about dying.
From what I read, this is what the Gautama Buddha meant with Anatta[1].
From an Islamic religious perspective, the soul is taken out of the body when you sleep and then put back into it when you wake up. Much the same way as when you die.
I try to be good to the guy who wakes up with my memories and body the next morning. That makes me no less unhappy about dying.