  We present the results from an observing campaign to confirm the peculiar
motion of the supermassive black hole (SMBH) in J0437+2456 first reported in
Pesce et al. (2018). Deep observations with the Arecibo Observatory have
yielded a detection of neutral hydrogen (HI) emission, from which we measure a
recession velocity of 4910 km s$^{-1}$ for the galaxy as a whole. We have also
obtained near-infrared integral field spectroscopic observations of the
galactic nucleus with the Gemini North telescope, yielding spatially resolved
stellar and gas kinematics with a central velocity at the innermost radii
($0.1^{\prime \prime} \approx 34$ pc) of 4860 km s$^{-1}$. Both measurements
differ significantly from the $\sim$4810 km s$^{-1}$ H$_2$O megamaser velocity
of the SMBH, supporting the prior indications of a velocity offset between the
SMBH and its host galaxy. However, the two measurements also differ
significantly from one another, and the galaxy as a whole exhibits a complex
velocity structure that implies the system has recently been dynamically
disturbed. These results make it clear that the SMBH is not at rest with
respect to the systemic velocity of the galaxy, though the specific nature of
the mobile SMBH -- i.e., whether it traces an ongoing galaxy merger, a binary
black hole system, or a gravitational wave recoil event -- remains unclear.
