The moduli spaces of Higgs bundles and holomorphic connections both have important affine holomorphic Lagrangian subvarieties, these are the Hitchin section and the space of opers, respectively. Both of these spaces arise from the same Lie theoretic mechanism, namely a regular nilpotent element of a Lie algebra. In this talk we will generalize these parameterizations to other nilpotents. The resulting objects are not related by the nonabelian Hodge correspondence, but by an operation called the conformal limit. Time permitting, we will also discuss their relation to Higher Teichmuller spaces.