  Laser pulses traveling through a plasma can feature group velocities
significantly differing from the speed of light in vacuum. This modifies the
well-known Volkov states of an electron inside a strong laser-field from the
vacuum case and consequently all quantum electrodynamical effects triggered by
the electron. Here we present an in-depth study of the basic process of photon
emission by an electron scattered from an intense short laser pulse inside a
plasma, labeled nonlinear Compton scattering, based on modified Volkov
solutions derived from first principles. Consequences of the nonlinear,
plasma-dressed laser dispersion on the Compton spectra of emitted photons and
implications for high-intensity laser-plasma experiments are pointed out.
