What can one say about the relation between the Hodge decompositions 
of $H^\\*(X,C)$ and $H^{\*}(X\_\sigma,C)$
for a complex algebraic smooth projective variety $X$ and $\sigma$ an automorphism
of the field of complex numbers ? I am also interested in conjectural results, e.g.
whether Standard Conjectures imply something about the relationship of
the Hodge structure of conjugate varieties $X$ and $X\_\sigma$.


For example, it appears that the Hodge numbers $h^{ab}=dim\_C H^a(X,\Omega\_X^b)$
and $h\_\sigma ^{ab}=dim\_C H^a(X\_\sigma,\Omega\_{X\_\sigma}^b)$ are necessarily
the same (being defined via de Rham cohomology); is this right (cf. 
discussion in the comments)? 
what other invariants/subspaces etc are the same or known to be possibly different ? 
Can one define a related Galois (or $Aut(C/Q)$) action on the Hodge structures,
something like where an automorphism $\sigma\in Aut(C/Q)$ takes the Hodge structure of $X$ 
into that of $X\_\sigma$'s ? 


I know of several references that show that conjugate varieties
may be rather different topologically. F.Charles, [Conjugate varieties with distinct real cohomology algebras](http://arXiv.org/abs/0706.3674v3)
lists most references I know of and well, proves what's in the title.
Serre constructed two two non-homotopic conjugated varieties with different fundamental groups;
his proof exploits the fact that for a CM elliptic curve, its fundamental group has different
$EndE$-module structures for different embeddings. [Abelson](http://www.maths.ed.ac.uk/~aar/topology/Topology%20Vol%2013/Abelson_Topologically-distinct-conjugate-varieties-with-finite-fundamental-group_1974.pdf) constructs two non-homotopic conjugated varieties with the same finite fundamental groups.



 comparing Hodge structures on cohomology of conjugate varieties
As Jason & Will have already commented, the Hodge numbers are the same for
conjugate pairs because of GAGA and the fact algebraic coherent cohomology behaves well with respect to field extensions. In fact, there would be an isomorphism of filtered vector spaces
$$(\mathbb{H}^i(X,\Omega^\bullet), F^\bullet)\otimes\_\mathbb{C} \mathbb{C}\_\sigma
\cong (\mathbb{H}^i(X\_\sigma,\Omega^\bullet), F^\bullet)$$
Nevertheless, the Hodge *structures* need not be the same because this may not be compatible with integral structures. 
For example, if $X$ is an elliptic curve with transcendental $j$-invariant, we
can find an automorphism with $j(X)\not= j(X\_\sigma)$ so that $H^1(X)\not\cong H^1(X\_\sigma)$ as Hodge structures. (So assuming I've understood your last question, that would be a no.)


