The question is basically in the title, but I want to make it more precise:
Given an oriented Riemannian 2-manifold $\Sigma$ one can take a quotient of the set $$ \mathcal{M}_+(\Sigma)=\{~c~\mid (\Sigma,c)\text{ is a Riemann surface for a given orientation on }\Sigma\} $$ by the action of $$ \text{Diff}_+(\Sigma)=\{~f~\mid~f:\Sigma\to\Sigma\text{ is an oriantation-preserving diffeo}\} $$ via $$ \mathcal{M}_+(\Sigma)\times\text{Diff}_+(\Sigma)\to\mathcal{M}_+(\Sigma),~(f,c)\mapsto f^*c $$ to obtain the moduli space $$ M(\Sigma)=\mathcal{M}_+(\Sigma)/\text{Diff}_+(\Sigma). $$ I understand that one aims to understand (some of) the structure of $\Sigma$ by studying $M(\Sigma)$. But doesn't the space $\mathcal{M}(\Sigma)/\text{Diff}(\Sigma)$ contain the same information on $\Sigma$ (without the subscripts they are just the set of all complex structures/group of all diffeo's)?
As far as I understand, for any complex structure $c$ there exists precisely one complex structure $\bar{c}$ such that $\text{id}:\Sigma\to\Sigma$, regarded as $\text{id}:(\Sigma,\bar{c})\to(\Sigma,c)$, is antiholimorphic. But this somehow makes the distinction between $c$ and $\bar{c}$ obsolete and a more 'natural' way to go would be to identify them as well and study $\mathcal{M}(\Sigma)/\text{Diff}(\Sigma)$ instead. Or is this somehow an 'uglier' space?
Google gave me some reference that in some case the moduli space consists of two connected components exchanged by complex conjugation, suggesting that this is generally not the case (however, lacking experience on the topic I found that article hard to read and I don't understand the precise setting). Also I found a reference which aims to generalize the subject to 'Klein surfaces' (for non-orientable Riemannian 2-manifolds), in which the moduli space is indeed $\mathcal{M}(\Sigma)/\text{Diff}(\Sigma)$ for the sake of having no $\text{Diff}_+(\Sigma)$. At third I found this SE-answer but this doesn't really help me any further - it merely points on the connection to the Teichmüller space but then the question arises again.
I lack intuition in the subject and would appreciate if anybody could tell me why we do not identify a complex structure $c$ with its 'conjugate' $\bar{c}$.