In contrast to the case of CMC tori, the space of compact higher genus CMC surfaces in the 3-sphere is unknown territory. For CMC tori the theory of Whitham deformations indicates that their moduli space is a connected set which is 1-dimensional at its smooth points. As a consequence it is possible to construct new CMC tori via Whitham deformations of known examples. This talk will describe experimental evidence suggesting the existence of generalized Whitham deformations for compact higher genus CMC surfaces. We will explain the integrable systems theory behind those deformations which is based on our recent spectral curve approach to higher genus minimal surfaces. Experiments reveal a behavior similar to that of tori. In particular, we discover a Delaunay type family of compact CMC surfaces of genus 2 which contains the Lawson genus 2 minimal surface.