I read part of it some time ago, but I just can't find it again;
It started about a girl who met an introverted classmate who saw humans as actual robots. The series takes a huge turn when the protagonist gets her "robot" arm fixed with cell phone parts after she was attacked. This eventually enables her to talk to herself in alternative realities, and eventually world/time travel. She does this with the goal of preventing the death of the girl who saw everyone as robots.

Kinda crazy story (but hey, it's manga) and I'm leaving out lots of detail here (the story switches from school life to murder mystery to some psychological world-traveling extravaganza)

It should be Qualia the Purple but I'm not completely sure.

It was! Thanks a lot! I was searching for world travel and time travel, but it was labeled under parallel dimensions.

(I'm now a bit disappointed to see that it was last updated 471 days ago).
Edit: Oh, there had been some updates, and it is actually finished now!

Last edited by Micket at 4:41 pm, Dec 28 2014