= Solution
Write $w$ for the complex coordinate. The inverse <stereographic projection> is
$$
\pi^{-1}(w)=\frac{(2\operatorname{Re}w,\,2\operatorname{Im}w,\,|w|^2-1)}{1+|w|^2},
$$
with $\infty$ mapped to the north pole. A rotation $R_z(\theta)$ about the third axis sends $w$ to $e^{i\theta}w$, represented by
$$
U_z(\theta)=\begin{pmatrix}e^{i\theta/2}&0\\0&e^{-i\theta/2}\end{pmatrix}\in SU(2).
$$
For the rotation $R_y(\theta):(x,y,z)\mapsto(x\cos\theta+z\sin\theta,y,-x\sin\theta+z\cos\theta)$, substitution in the <stereographic projection> gives
$$
w\longmapsto\frac{cw-s}{sw+c},\qquad
U_y(\theta)=\begin{pmatrix}c&-s\\s&c\end{pmatrix}\in SU(2),\quad
c=\cos(\theta/2),\ s=\sin(\theta/2).
$$
Both displayed matrices are unitary with determinant one, hence belong to the <special unitary group>.
Use the rotation-generation result that every $R\in SO(3)$ can be written $R_z(\alpha)R_y(\beta)R_z(\gamma)$, with angles chosen suitably. This is the Euler-angle decomposition, which is allowed here. Composition of <Möbius transformations> corresponds to multiplication of their matrix representatives, so
$$
\boxed{\pi R\pi^{-1}\text{ is represented by }U_z(\alpha)U_y(\beta)U_z(\gamma)\in SU(2).}
$$
The formulas extend at poles in the <Riemann sphere>. This realizes <sphere rotations as special-unitary Möbius transformations>. The two representatives $U$ and $-U$ define the same <Möbius transformation>, as expected for the double covering of the <special orthogonal group>.
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