The moduli space of (non-singular) Higgs bundles may be seen as either the holomorphic symplectic quotient or hyperkahler quotient of an infinite dimensional hyperkahler affine space. The moduli space of all parabolic Higgs bundles is a holomorphic Poisson reduction of a similar space, but it is not clear what is now analogous to the hyperkahler reduction. We discuss work in progress where we try to answer this question. This will shed light on recent work where we generalized the hyperholomorphic line bundle of Haydys and Hitchin to a more general class of hyperkahler manifolds. We also discuss a candidate in this set-up for the analog of the $L^2$-norm of a Higgs field, which has played an important role in studying the topology of moduli spaces of Higgs bundles.