[Nadya]

This is the exact explanation used in the anime Steins;Gate. Okabe (the main character) is capable of 'moving timelines' which is treated as jumping between different parallel worlds.

He leaves his friends behind in a world where WW3 breaks out over world powers fighting over research for a working time machine. In a way, he is leaving his friends to die in a post-apocalyptic WW3...but that doesn't matter because he's saving them in his 'current' timeline. The other worlds, from his perspective, "don't exist" anymore.

>In all actuality, "you" died and another "you" took your place. But to you and everyone around you, it seems like nothing has changed, so you carry on with your life.

In the show they call this power "Reading Steiner" when he "replaces himself" in a new timeline. Since he is capable of remembering what happened in the other timelines (nobody else does - as from their perspective these other timelines never existed).

Time travel is fun to think about. :P