In analogy to Simpson's complex non-abelian Hodge correspondence, the conjectural p-adic Simpson correspondence aims to relate p-adic representations of the étale fundamental group of a smooth proper rigid space X over C_p to Higgs bundles on X. I will discuss how this correspondence can be naturally rephrased as studying vector bundles on the pro-étale site of X. Based on this reformulation, I will explain how one can define p-adic analytic moduli spaces of representations and Higgs bundles, and how these give rise to a new geometric perspective on the p-adic Simpson correspondence in the case of curves. Surprisingly, in some respect, this seems to be better behaved than what one encounters in the complex theory.