Just a quick question about naming: in complex analysis, it appears to me that an "Automorphism" of $\Omega\subseteq \mathbb C$ means a conformal mapping from a subset $\Omega$ to itself. So, an automorphism doesn't necessarily preserve the group structure of $\mathbb C$ However, in group theory, "automorphisms" should be isomorphisms.
Why the same word mean such two different things?
NOTE Someone appears to say that this is a duplicate question, so let me make it more clear: My question is, does the word "automorphisms" mean means completely different things in group theory and complex analysis? Or actually they mean similar things?
That question does not mention groups at all.
This is because the notion of isomorphism depends on the category in which you are working. If we are working in the category of groups, then the right maps are those which preserve the structure of the group: that is, maps of groups $f:G\to H$ so that $f(xy)=f(x)f(y)$ for $x,y\in G$. This is the notion of a group homomorphism. Then, an isomorphism of groups should be a bijective homomorphism with homomorphic inverse.
The real idea behind the scenes is that an isomorphism (of groups) is equivalent to a pair of maps $\phi:G\to H$ and $\psi:H\to G$ which are structure preserving in the sense above with $\phi\circ \psi=\mathbb{1}_H$ and $\psi\circ \phi =\mathbb{1}_G$. So, an automorphism is a group homomorphism $\phi:G\to G$ which is a bijection with a homomorphism for inverse.
If we change to the complex case, then the thing that changes is the notion of "structure preserving." If $\Omega\subseteq \mathbb{C}$ is open, it is in particular a $1-$dimensional complex manifold, and is equipped with a complex structure $\mathcal{O}$. So, maps between such spaces should be "structure preserving" also. The appropriate notion of such maps is a holomorphic map. So, an isomorphism of subdomains of the complex plane is a pair of such maps as above, $f:\Omega_1\to \Omega_2$ and $g:\Omega_2\to \Omega_1$ holomorphic with $f\circ g=\mathbb{1}_{\Omega_2}$ and $g\circ f=\mathbb{1}_{\Omega_1}$.
An automorphism is then a map $f:\Omega\to \Omega$ which is a holomorphism with holomorphic inverse.
So, these notions of automorphism/isomorphism are examples of the same more general idea.
As for why an isomorphism of complex subdomains need not preserve group structure: well, firstly notice that a subdomain of the complex plane is almost never equipped with a natural group structure. However, a complex structure need not be anything to do with a group structure. More precisely, the complex structure $\mathcal{O}_\mathbb{C}$ of $\mathbb{C}$ makes no mention of the group structure of $\mathbb{C}$, so we should not expect that an isomorphism of complex spaces would pay any heed to the group structure.