  We present relativistic, radiation magnetohydrodynamic simulations of
supercritical neutron star accretion columns in Cartesian geometry, including
temperature-dependent, polarization-averaged Rosseland mean opacities
accounting for classical electron scattering in a magnetic field. Just as in
our previous pure Thomson scattering simulations, vertical oscillations of the
accretion shock and horizontally propagating entropy waves (photon bubbles) are
present in all our simulations. However, at high magnetic fields
$\gtrsim10^{12}$~G, the magnetic opacities produce significant differences in
the overall structure and dynamics of the column. At fixed accretion rate,
increasing the magnetic field strength results in a shorter accretion column,
despite the fact that the overall opacity within the column is larger.
Moreover, the vertical oscillation amplitude of the column is reduced.
Increasing the accretion rate at high magnetic fields restores the height of
the column. However, a new, slower instability takes place at these field
strengths because they are in a regime where the opacity increases with
temperature. This instability causes both the average height of the column and
the oscillation amplitude to substantially increase on a time scale of
$\sim10$~ms. We provide physical explanations for these results, and discuss
their implications for the observed properties of these columns, including
mixed fan-beam/pencil-beam emission patterns caused by the oscillations.
