Why is quaternion algebra 4D and not 3D? Complex algebra is 2D and what is known as quaternion algebra jumps to 4D.
$ i^2 = j^2 = k^2 = ijk = -1 $
Using $1, i, j,$ and $k$ as the base (where complex uses $1$ and $i$ (or $j$ if you are an EE)) which results in a 4-axis space. Why is there no 3D algebra?
There are only four normed division algebras (algebras where division by nonzero elements is possible) over the reals: the reals themselves, the complex numbers, quaternions, and a strange (alternative but nonassociative) algebra called octonions.
The reason that the dimensions are in geometric progression 1, 2, 4, 8 is that they can be derived from repeatedly applying the Cayley-Dickson construction, which doubles the dimension at each step. This explains the absence of dimension 3.
Generally, as you go up or down the Cayley-Dickson ladder you lose properties (as well as gaining some properties). From the reals to the complex numbers you lose order; going to the quaternions you lose commutativity; going to the octonions you lose associativity; going to the sedonions you're no longer alternative or a division algebra.