Your question is a very good one, and one that has attracted the attention of mathematicians for a number of years. As it happens, there is exactly one other dimension for which there is something like the multiplication that exists for real, complex, and quaternionic numbers. It was invented by Cayley and it is called "the octonions" by some people since each "number" has eight coordinates. You can think of an octonion as a pair of quaternions roughly in the same way that a quaternion can be represented as a pair of complex numbers and a complex number as a pair of reals. There is something very special about the dimensions of these numbers that translates into facts about finding a global coordinate system on the spheres of dimension one lower, a so-called "parallelization". The 1-,3-,and 7-dimensional spheres are all "parallelizable" and the others aren't. It's subtle.
It is true that quite a few other properties of vectors in 2- and 3-space generalize to n-space very nicely, and that provides the material for linear algebra classes. Maybe I can survey that in class today or next week.