  We present a detailed characterization of 2MASS J04435750+3723031, a low-mass
companion orbiting the young M2 star, 2MASS J04435686+3723033, at 7.6
arcseconds (550 AU) with potential membership in the 23 Myr $\beta$ Pictoris
moving group ($\beta$PMG). Using near-infrared spectroscopy of the companion
from IRTF/SpeX we have found a spectral type of M6 $\pm$ 1 and indications of
youth through age-sensitive absorption lines and a low surface gravity index
(VL-G). A young age is supported by H$\alpha$ emission and lithium absorption
in the host. We re-evaluate the membership of this system and find that it is a
marginally consistent kinematic match to the $\beta$PMG using $Gaia$ parallaxes
and new radial velocities for the host and companion. If this system does
belong to the $\beta$PMG, it would be a kinematic outlier and the companion
would be over-luminous compared to other similar ultracool objects like PZ Tel
B; this would suggest 2M0443+3723 B could be a close brown dwarf binary
($\approx$52+52 M$_\mathrm{Jup}$ if equal-flux, compared with 99 $\pm$ 5
M$_\mathrm{Jup}$ if single), and would make it the sixth substellar companion
in this group. To test this hypothesis, we acquired NIR AO images with Keck
II/NIRC2, but they do not resolve the companion to be a binary down to the
diffraction limit of $\sim$3 AU. If 2M0443+3723 AB does not belong to any
moving group then its age is more uncertain. In this case it is still young
($\lesssim$30 Myr), and the implied mass of the companion would be between
$\sim$30--110 M$_\mathrm{Jup}$.
