Steins;Gate, Amnesia Recommendations
Parallel worlds, mystery,... very similar, if you like amnesia you'll like steins gate too. XD
Both are about time travel, though the principle through which they occur are slightly different.
Also, when the time travel happens, events become slightly different. The protagonist and one other person also retains their memories from the previous times they "travelled" back. In steins;gate, this is shown in the World-line shifts when d-mails are sent. (not when okabe actually travels back further in time), and in Amnesia, it just happens when the main character is in some sort of danger.
Though, Steins;Gate is more content-driven than Amnesia.
If you like stories where people move to different worlds and travel through time over and over again to save a dear friend, where the plot takes unexpected twists and turns, then Steins;Gate and Amnesia are exactly what you're looking for.
*side note Steins;Gate is the more mature and plot-driven one out of the two.
They both have parallel timelines involved, the main-characters have to re-live a certain time more than once, each time getting involved with one of their friends. Both are partly harem-type.
Steins Gate is about a young man, who tries finding the way to save his friend.
In Amnesia the main-character girl does not have the control when to retry, and she does not wish to reach a certain ending. Instead, it seems to have a mystic plot.