In this paper, we discuss the possibility of using x-ray Compton scattering as a probe of the outer electronic structure of ions immersed in warm dense matter. It is proposed that the x-ray free-electron lasers currently under construction will provide an ideal tool for this, with the main pulse being used to create a uniform well-defined sample and the third harmonic providing a clean monochromatic probe. We model the plasma photon scatter spectrum by combining self-consistent finite-temperature electronic structure calculations with molecular dynamics simulations of the ion-ion structure factor. In particular, we present bound-free Compton profiles that are more accurate that those obtained using form factor or impulse approximations.