Riemann sphere in hypercomplex number sets?

Question

Do Riemann spheres work on bigger number sets than complex numbers? I think it would be interesting to think of quaternions as a 5D sphere or something, but I really don't know if the maths of Riemann spheres work in other dimensions. Thanks.

Answer 1

Yes. These correspondences are actually really important when trying to understand the topology of spheres because of the existence of the corresponding Hopf maps. Just like how $S^2$ is homeomorphic to the complex projective line $\mathbf{CP}^1$, we have the correspondences $$\begin{align}S^1&\cong\mathbf{RP}^1\\ S^2&\cong\mathbf{CP}^1\\ S^4&\cong\mathbf{HP}^1\\ S^8&\cong\mathbf{OP}^1 \end{align}$$ for the real, complex, quaternionic, and octonionic number systems, respectively.