There are all sorts of variants of the answer of abx and of the example in the question:

By Hitchin's self-duality paper, you can identify the $SL(2,\mathbb C)$-representation space with the moduli space of Higgs bundles (holomorphic bundles together with Higgs fields). Note that in this case, the spaces are only isomorphic as real analytic varieties, but the corresponding complex structures differ. In fact, one gets a hyper-kaehler structure from the two different complex structures.

As another example, you can consider a punctured Riemann surface $\tilde M=M\setminus \{p_1,..,p_n\}$ and consider the subspace of the representation space
$Hom(\pi_1(\tilde M),SU(2))/SU(2)$
given by fixed local conjugacy classes $C_i$ around $p_i.$ Then, this space is isomorphic (as a real analytic variety) to the moduli space of parabolic structures on $\tilde M,$ where the parabolic weights are determined by $C_i.$ Generalizing to $SL(2,\mathbb C)$ gives the moduli space of parabolic Higgs bundles.

There is another example in Hitchin's paper on the self-duality equations, namely the Teichmüller space, which can be identified with the space of certain Fuchsian representations of the fundamental group of the surface modulo conjugation. On the other hand, from Hitchin, this space can be identified with a subspace of the moduli space of Higgs bundles.