Is my claim true? If so, is my "proof" correct?
Claim: All rotations about a point in the complex plane also Möbius maps. Proof: We have shown that the set of Möbius maps is a group. So we can do the following scaling, translations, and inversions.
\begin{equation*} \begin{aligned} f(z+k) & =\frac{az+(ak+b)}{cz+(ck+d)} \text{ where } a(ck+d)-c(ak+b)=ad-bc \neq 0 \\ f(kz) & = \frac{akz+b}{ckz+d} \text{ where } (ak)d-b(ck)=k(ad-bc) \neq 0 \text{ if } k \neq 0 \\ f(\frac{1}{z})=\frac{bz+a}{dz+c} \text{ where } bc-ad \neq 0 \end{aligned} \end{equation*} Hence Möbius transformations composed with these generators yield further Möbius transformations and the result follows.