Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some moduli spaces, that correspond to Betti and de Rham cohomologies. For a compact curve (as far as I understand), this construction indeed gives a mixed Hodge structure on the formal functions near a trivial connection coming from the one on the fundamental group.
For noncompact curves, corresponding moduli spaces are described in "Harmonic bundles on noncompact curves" by C.Simpson. Is there any explanation, why this definition is the right one? Is it possible to extract from this structure some information about the fundamental group of the noncompact curves? Say, for a projective line with punctures?