I am trying to understand (Saito's?) category of mixed Hodge modules as a category (i.e. I am not interested in its construction, just in properties of objects and morphisms). I would be grateful for any nice references for this (though I have some texts already); yet currently I am interested in the following question.
For algebraic varieties over a subfield $k$ of the field of complex numbers, can one define certain mixed Hodge modules with some $k$-structure that would be related with the $k$-structure on the De Rham cohomology of $k$-varieties? Please tell me also if such a structure could be obtained from the usual 'complex' Hodge modules, or if the presence of a $k$-structure 'does not affect morphisms significantly'.