The question is already in the heading. Determine all conjugacy-classes of the unit-quaternions $S^3=\{h \in \mathbb{H} | \: \bar{h}\cdot h = 1\}$.
My only idea to tackle this question would be:
Since $S^3$ is compact and the action (conjugation) is smooth, the conjugacy classes should be compact embedded submfs.
Another idea would be to somehow use the representation here: The conjugation action $\mathbb{H}^*\times \mathbb{H} \rightarrow \mathbb{H}$ restricted to unit-quaternions yields an orthogonal representation
Any help would be appreciated
Remember all unit quaternions have polar forms $e^{\theta\mathbf{u}}$ where $\theta$ is a convex angle (i.e. $0\le\theta\le\pi$) and $\mathbf{u}$ is a unit vector (so, a sqrt of $-1$), and Euler's formula says $e^{\theta\mathbf{u}}=\cos\theta+\sin\theta\,\mathbf{u}$. Conjugating by $e^{\theta\mathbf{u}}$ has no effect on reals but has the effect of rotating vectors by $2\theta$ around the $\mathbf{u}$-axis. Any vector of a given size can be rotated to any other vector of the same size.
Thus, for each $0\le\theta\le\pi$ there is a conjugacy class of all $e^{\theta\mathbf{v}}$ with $\mathbf{v}\in S^2$. For $\theta=0,\pi$ these are singletons $\{+1\},\{-1\}$ respectively but otherwise they are two-spheres within $S^3$.