  We investigate the effect of spin-orbit coupling on the in-gap bound states
localized at magnetic impurities in multi-band superconductors with
unconventional (sign-changed) and conventional (sign-unchanged) $s$-wave
pairing symmetry, which may be relevant to iron-based superconductors. Without
spin-orbit coupling, for spin-singlet superconductors it is known that such
bound states cross zero energy at a critical value of the impurity scattering
strength and acquire a finite spin-polarization. Moreover, the degenerate,
spin-polarized, zero energy bound states are unstable to applied Zeeman fields
as well as deviation of the impurity scattering strength away from criticality.
Using a T-matrix formalism as well as analytical arguments, we show that, in
the presence of spin-orbit coupling, the zero-energy bound states localized at
magnetic impurities in unconventional, sign-changed, $s$-wave superconductors
acquire surprising robustness to applied Zeeman fields and variation in the
impurity scattering strength, an effect which is absent in the conventional,
sign-unchanged, $s$-wave superconductors. Given that the iron-based multi-band
superconductors may possess a substantial spin-orbit coupling as seen in recent
experiments, our results may provide one possible explanation to the recent
observation of surprisingly robust zero bias scanning tunneling microscope
peaks localized at magnetic impurities in iron-based superconductors provided
the order parameter symmetry is sign changing $s_{+-}$-wave.
