  Brownian motion of a mobile impurity in a bath is affected by spin-orbit
coupling (SOC). Here, we discuss a Caldeira-Leggett-type model that can be used
to propose and interpret quantum simulators of this problem in cold Bose gases.
First, we derive a master equation that describes the model and explore it in a
one-dimensional (1D) setting. To validate the standard assumptions needed for
our derivation, we analyze available experimental data without SOC; as a
byproduct, this analysis suggests that the quench dynamics of the impurity is
beyond the 1D Bose-polaron approach at temperatures currently accessible in a
cold-atom laboratory -- motion of the impurity is mainly driven by dissipation.
For systems with SOC, we demonstrate that 1D spin-orbit coupling can be 'gauged
out' even in the presence of dissipation -- the information about SOC is
incorporated in the initial conditions. Observables sensitive to this
information (such as spin densities) can be used to study formation of steady
spin polarization domains during quench dynamics.
