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.