I recently found this extremely intriguing video game, Miegakure, in which you solve puzzles using the 4th dimension. http://miegakure.com/
This game is completely mind boggling. The player has to switch between the 3rd dimensions and 4th dimensions to do absurd tasks like walking "through" walls. At first I thought you were just going into another screen where the wall does not exist, walking past where the wall was, and then switching back to the 3rd dimension. However, the more I read about the creation of this game, the more I'm convinced it's not that.
The creator literally programmed the game in 4th dimensional code; the coordinates are sort of like (x,y,z,w). From what I have figured out through reading, the 4th dimension is similar to how we view the differences between the 2nd and 3rd dimensions. Every 3d shape has 2d faces, so therefore every 4d shape has 3d faces - still impossible to comprehend to be honest. I do know that we could do seemingly ridiculous things if were able to move about in 4 dimensions.
For instance, if you were to draw a simple square on a piece of paper, you could see everything that is "inside" the square. The same concept could be true if we could visualize the 4 dimensionally. We could see what is inside of 3d objects by moving to another facet of it. Also, it would seem like you could teleport around by moving through the 4th dimension.
All of these are possible theories and ideas, but they seem to work. The physical feats we could accomplish would be amazing, if only we could move about 4 dimensionally.