  Massive star-forming galaxies in the high-redshift universe host large
reservoirs of cold gas in their circumgalactic medium (CGM). Traditionally,
these reservoirs have been linked to diffuse H I Lyman-$\alpha$ (Ly$\alpha)$
emission extending beyond $\approx 10$ kpc scales. In recent years,
millimeter/submillimeter observations are starting to identify even colder gas
in the CGM through molecular and/or atomic tracers such as the [C II]
$158\,\mu$m transition. In this context, we study the well-known J1000+0234
system at $z=4.54$ that hosts a massive dusty star-forming galaxy (DSFG), a
UV-bright companion, and a Ly$\alpha$ blob. We combine new ALMA [C II] line
observations taken by the CRISTAL survey with data from previous programs
targeting the J1000+0234 system, and achieve a deep view into a DSFG and its
rich environment at a 0.2" resolution. We identify an elongated [C II]-emitting
structure with a projected size of 15 kpc stemming from the bright DSFG at the
center of the field, with no clear counterpart at any other wavelength. The
plume is oriented $\approx 40^{\circ}$ away from the minor axis of the DSFG,
and shows significant spatial variation of its spectral parameters. In
particular, the [C II] emission shifts from 180 km/s to 400 km/s between the
bottom and top of the plume, relative to the DSFG's systemic velocity. At the
same time, the line width starts at 400-600 km/s but narrows down to 190 km/s
at top end of the plume. We discuss four possible scenarios to interpret the [C
II] plume: a conical outflow, a cold accretion stream, ram pressure stripping,
and gravitational interactions. While we cannot strongly rule out any of these
with the available data, we disfavor the ram pressure stripping scenario due to
the requirement of special hydrodynamic conditions.
