Hyper-Dimensional Brainfuck is an infite-dimensional version of the Brainfuck programming language. Instead of a tape, we have a "world." Rays parallel to the axis of each dimension in the world are called "vectors," and act very much like the Brainfuck tape. You can switch between vectors using the ^ and V commands.

+ - increase current cell
- - decrease current cell
> - shift positively along current vector
< - shift negatively along current vector
^ - switch to a vector parallel to the next dimensional axis, that pass through the current cell
V - switch to a vector parallel to the previous dimension axis, that pass through the current cell
. - print out contents of current cell
, - input from STDIN
[ - skip to matching ] if cell values zero
] - return to matching [

Let's say we're only considering a 2-dimensional plane of the world

[0,1][1,1]
[0,0][1,0]

You start at 0,0. To shift to 1,0, you would use >. To switch to 1,1 from there, you would use ^>. From there, to switch to 0,1, you would just use V<, and to switch back to 0,0, you use ^<

Now let's consider a three dimensional cube, specifically a 3x3x3 cube.
To go from the bottom, left, front corner of the cube, i.e (0,0,0) to the top, right, back corner, i.e. (2,2,2) You would do the following:

>>^>>^>>

For a 4x4x4x4 tesseract:
>>>^>>>^>>>^>>>