Only if there's an achievement for killing your past self. And anothsrfor killing your future self in the tutorial and then completing the game until the loop back (at which point your past self shoots you in the face).
As it is, game time travel plots rarely work that well. I kind of have up on The Legend of Zelda: Das Fluten Tuten after discovering that after every main dungeon I'd have to go to a specific point in the game world, travel back in time, do stuff to unlock the next dungeon, go back to that point, go forwards in time, and then I could actually unlock the dungeon. At least Oracle of Ages spread time travel points around the map, and eventually let you time travel anywhere. It's really best when it's mostly an excuse to mash level themes together.