Marco,
Since I don't fully understand what you are asking, let me instead discuss
what you say is motivating you. If it isn't relevant, you can say so.
When $X$ is a compact Kaehler manifold
then what Simpson calls the Betti moduli space is $M_B(X)$
is the set of semisimple representations in $Hom(\pi_1(X),GL_n(\mathbb{C}))/conjugacy$.
This can be made into a variety (actually a scheme). When $n=1$, this is simply
$$Hom(\pi_1(X),\mathbb{C}^*)=H^1(X,\mathbb{C}^*)$$
On the other side, the "Dolbeault moduli space" $M_{Dol}(X)$ is the moduli space of Higgs bundles (satisfying appropriate
conditions). Again, in the rank one case, this simply the cotangent bundle
of the Picard torus $T^*Pic^0(X)= Pic^0(X)\times H^0(X,\Omega_X^1)$. The correspondence
in this case can be deduced from standard Hodge theory; it
is sketched on page 21 of Simpson's Higgs bundles and local systems, and it
is worthwhile to fill in the details yourself.
One thing you will notice is that this story doesn't really have much to do with cohomology beyond $H^1$. Actually, this isn't quite true. Even though the FAQ discourages discussion of open problems, let me mention one that may have some relevance to your question:
Is the image of the natural map $H^*(\pi_1(X),\mathbb{C})\to H^*(X,\mathbb{C})$ a sub Hodge structure?
