I can't see how anything more than 3 dimensions could be of any use beyond maybe a novelty. If you consider time to be a dimension, then that might be useful in a game, but a 4th/5th/etc. spatial dimension would be useless.
Maybe it would work if the camera was restricted to the XYZ region (as with normal 3D games) and you were allowed to "sidestep" into the 4th dimension to go "around" walls. But then what would be in the parallel 3D space? A separate world? A slightly different world? A blank world? Really, more than 3 dimensions really don't work well unless you're referring to time.
This kind of this is best done as a "fictional" addition, where you artistically create disorienting effects. Have you ever tried to imagine a full 4D world? It's not easy. A 4D game would look like nothing more than weirdly warping lines which resemble nothing at all, since mapping a 4D world onto a 2D screen would be like mapping a 3D world onto a 1D screen.
A haunted house in 3D wouldn't make sense since a house is a human 3D object. You would have to "invent" the rest of the 4-dimensional shape, where the 3D house is only a single "slice" out of a 4D world.
A 4D space warp in a 3D space game would make a little more sense, where possibly your weapons could warp 3D space to open up wormholes or teleport objects.
Maybe it would work if the camera was restricted to the XYZ region (as with normal 3D games) and you were allowed to "sidestep" into the 4th dimension to go "around" walls. But then what would be in the parallel 3D space? A separate world? A slightly different world? A blank world? Really, more than 3 dimensions really don't work well unless you're referring to time.
I actually see a lot of use for a multiple dimension engine(beyond 3). Think about how awesome it would be for a spooky video game to have a haunted house that exists in 4-dimensional space. Or in a sci-fi space ship game you can have some sort of ray gun that temporarily makes part of the arena more than 3 spacial dimensions(totally disorientating players in online games!)
This kind of this is best done as a "fictional" addition, where you artistically create disorienting effects. Have you ever tried to imagine a full 4D world? It's not easy. A 4D game would look like nothing more than weirdly warping lines which resemble nothing at all, since mapping a 4D world onto a 2D screen would be like mapping a 3D world onto a 1D screen.
A haunted house in 3D wouldn't make sense since a house is a human 3D object. You would have to "invent" the rest of the 4-dimensional shape, where the 3D house is only a single "slice" out of a 4D world.
A 4D space warp in a 3D space game would make a little more sense, where possibly your weapons could warp 3D space to open up wormholes or teleport objects.