Just an idea I was toying with a few years ago, had a little bit of a prototype but not much so words are going to have to explain the mechanics of it.
The players viewpoint would most likely be first person perspective or even 3/4 top down perspective. Either would work with the concept.
Flow of Gameplay
The gameplay would take place over a series of maps or levels either indoor or outdoor in a terrestrial setting, perhaps starting modern times. Each map or level would represent a different point in time, and entry to the level except for levels in the modern time would be made via a time machine.
Each level would be played out of time sequence with a number of level puzzles or challenges. At some point into the game, the command structure requires you to use the time machine to fix problems created by your initial tampering in that time. The risk is that you are not allowed to cross paths with the previous "you", that is that you cannot cross the viewpoint of your previous self on that level. If you do, you are wiped out and must begin the process on that level from your first attempt at the level all over again.
The concept is a little bit similar to the movies Back To The Future 2 and Millenium, which involved time paradoxes effecting time spaces as sort of earth quakes or obliteration of the violators of the laws of time/space. The game itself would borrow nothing from the concept of the movie except for time travel which is a popular theme in fiction and movies but has rarely been explored in video games.
Eventually the player would end up dealing with as many as three or possibly four instances of themselves within each level, maybe with a missions like being required to "disable" one of the prior instances of yourself to prevent it from completing its mission.
The concept could run on any existing modern 3d engine without too much modification, and the only requirement would be that the engine supported the recording of motion paths and events on the level, for later playback while the player entity played the level again.
The playback of a motion for any entities in the level would include the usual collision checks during the process of playback as the second instance of the player entity might collide with any any such entities on that level. As soon as a collistion was detected, the force and impact data would be applied from that point on and the entity that was collided with would be taken off of playback mode and interact regularly from therein.