This questions has been answered already: Exist domains in complex plane with only trivial automorphisms?
"Start with the "box" $\{ζ\in C∣|Reζ|<2,|Imζ|<2\}\{ζ\in C∣|Reζ|<2,|Imζ|<2\}$, remove the four closed disks of radius 0.1 and centered at $\pm 1\pm i$, and "perturb one of the holes" by 0.1."
but I was hoping to find a simpler example.
Attempt
Checking if the punctured annulus $A_{2}:=\mathbb{D}\setminus \{0,\frac{1}{2}\}$, is an example.
Since $f\in Aut(A_{2})$ is bounded by 1, we can extend to $\tilde{f}\in Aut(\mathbb{D}\setminus\{0\})$ and so $f=e^{i\theta}z$. But any rotation will land on the $\frac{1}{2}$ and so we must have $f=z$.
