I'm not sure that teleportation to two locations simultaneously breaks the concept of "me-ness". Neither of them is the "me" which existed prior to the teleportation, and only one of them is the "me" which is being asked the question post-teleport. They would certainly be very similar people, but their experiences diverge at the instant of reappearance.

The concept of "me" as "integral of experience over time" seems fairly robust, if rather difficult to express succinctly in any specific case...