Every biholomorphic self-map of the punctured plane is of the form az or a/z

If $z_n\to 0$ in ${\mathbb{C}}^{\times }$ and $f(z_n)\to a\in {\mathbb{C}}^{\times }$, continuity of $f^{-1}$ on ${\mathbb{C}}^{\times }$ forces $z_n=f^{-1}(f(z_n))\to f^{-1}(a)\in {\mathbb{C}}^{\times }$, a contradiction. Hence every cluster value of $f(z)$ as $z\to 0$ lies in $\{0,\mathrm{\infty }\}$. If both $0$ and $\mathrm{\infty }$ were cluster values, then every sufficiently small punctured disc would have image meeting both $\{\mathrm{\mid }w\mathrm{\mid }<1\}$ and $\{\mathrm{\mid }w\mathrm{\mid }>1\}$; connectedness of that image would then produce points with $\mathrm{\mid }f(z)\mathrm{\mid }=1$ approaching $0$, giving a finite nonzero cluster value after all. Therefore $f(z)$ tends to a single limit ${\mathrm{\ell }}_0\in \{0,\mathrm{\infty }\}$ as $z\to 0$. The same argument applied to $z\mapsto f(1\mathrm{/}z)$ shows that $f(z)$ tends to a single limit ${\mathrm{\ell }}_{\mathrm{\infty }}\in \{0,\mathrm{\infty }\}$ as $z\to \mathrm{\infty }$.

Let $F$ be the extension of $f$ to $\widehat{\mathbb{C}}$ with $F(0)={\mathrm{\ell }}_0$ and $F(\mathrm{\infty })={\mathrm{\ell }}_{\mathrm{\infty }}$, and let $G$ be the analogous extension of $f^{-1}$. Step 1.1 gives continuity of both extensions at the added points, and on the dense subset ${\mathbb{C}}^{\times }$ one has $G\circ F=\mathrm{id}=F\circ G$. By continuity the same identities hold on all of $\widehat{\mathbb{C}}$, so $F$ is a sphere homeomorphism and $F(\{0,\mathrm{\infty }\})=\{0,\mathrm{\infty }\}$.

Near each of $0$ and $\mathrm{\infty }$, the homeomorphism $F$ lands either in a bounded finite chart or in a neighbourhood of $\mathrm{\infty }$. In the first case the corresponding chart expression is bounded near the puncture and extends holomorphically by [L2]; in the second case its reciprocal is bounded and again extends holomorphically by [L2]. Thus $F$ is meromorphic at both added points, and therefore on the whole sphere.

Fact [L1] makes $F$ a rational sphere map of degree $1$, hence Möbius. A Möbius map preserving the set $\{0,\mathrm{\infty }\}$ is either $z\mapsto az$ or $z\mapsto a\mathrm{/}z$ with $a\mathrm{\ne }0$, and restricting back to ${\mathbb{C}}^{\times }$ gives exactly the claimed automorphisms.