Unit quaternion ($\mathrm{Sp}(1)\cong S^3$)

A quaternion of norm $1$ is a unit quaternion. They form the topological group $\mathrm{Sp}(1)$ and identify with the sphere $S^3\subset {\mathbb{R}}^4$. Multiplication and inversion preserve the norm. The Hurewicz theorem identifies ${\pi }_3(S^3)$ with $H_3(S^3;\mathbb{Z})$; hence homotopy-group addition in a topological group shows that $q\mapsto q^m$ has mapping degree $m$, rather than $m^2$.