The effect of a superstrong magnetic field of 10(13) G on Compton scattering for thermal photons with electrons is studied by the method of quantum field theory. The Stokes parameters for the scattered photon are computed explicitly in terms of the state of polarization of the incident wave, the electron-cyclotron frequency, the angle of incidence and the angle of scattering. In the nonrelativistic limit, our results reduce to the well known theory for magnetic Thomson scattering. The total scattering cross section derived from the Stokes parameter S-0 for the extraordinary X-mode in this case may be cast very roughly in the form sigma(B)approximate to(omega/omega(c))(2) sigma(0) where sigma(B) and alpha(0) respectively denote the Thomson cross section with and without the presence of the magnetic field, omega(c) represents the classical gyrofrequency for the electrons and omega is the frequency of the incident wave provided that the dependence on the angle of incidence is neglected. The criterion for the magnetic field to substantially affect the Stokes parameters is that the photon frequency be less than the electron-cyclotron frequency. For instance, the Thomson cross section is significantly reduced and the photon mean free path is greatly enhanced for photons with omega much less than omega(c). In the absence of the strong magnetic field, the differential and total cross sections (derived from the first Stokes parameter S-0) become the famous Klein-Nishina formula. Specific application to high energy astrophysics such as neutron star cooling and models for compact gamma-ray source are currently under investigation and will be briefly discussed.