I don't think that octonions are the obvious next step here. They are a non-associative algebra, and therefore incredibly difficult to deal with. Starting with

I don't think that octonions are the obvious next step here. They are a non-associative algebra, and therefore incredibly difficult to deal with. Starting with real n-space, one can form the Clifford algebra, which essentially gives a method of multiplying vectors which "knows" something about the length and angle of vectors. The even subalgebra of the Clifford algebra gives a very convenient way of encoding rotations on real n-space. Furthermore, the Clifford algebra is always associative, and works for any n. If you apply this construction for n=1, 2, 3, you get back the real numbers, complex numbers, and quaternions respectively. If you apply this for n=4, you get back an 8-dimensional associative algebra encoding rotations in 4-space.