There is an N-dimensional generalisation for complex numbers, quaternions and hypercomplex numbers called Clifford algebra. I'm not an expert in that field (coming from physics and optics) but apparently it's not by chance that quaternions are connected with the rotation group SO(3) and complex numbers with SO(2). You can generalize to SO(n) with Clifford algebra.
This is closely related to the exterior algebra, which is where bivectors live. But I don't know much about Clifford algebras either. I didn't know that they are more directly connected to Quaternions. Thanks!
The 'geometric product' mentioned in the article is just Clifford's product. The 'geometric algebra' is the same thing as the Clifford algebra induced by the scalar product.
