  Quasars at early redshifts ($z > 6$) with companion galaxies offer unique
insights into the growth and evolution of the first supermassive black holes.
Here, we report on a 150 ks Chandra observation of PSO J308.0416$-$21.2339, a
$z=6.23$ quasar with a merging companion galaxy identified in [C II] and
rest-frame UV emission. With $72.3^{+9.6}_{-8.6}$ net counts, we find that PSO
J308.0416$-$21.2339 is powerful ($L_X = 2.31^{+1.14}_{-0.76} \times 10^{45}\
\textrm{erg}\,\textrm{s}^{-1}\,\textrm{cm}^{-2}$ in rest-frame $2.0-10.0$ keV)
yet soft (spectral power-law index $\Gamma=2.39^{+0.37}_{-0.36}$ and
optical-to-X-ray slope $\alpha_{\rm OX} = -1.41 \pm 0.11$). In addition, we
detect three hard-energy photons $2.''0$ to the west of the main quasar,
cospatial with the brightest UV emission of the merging companion. As no
soft-energy photons are detected in the same area, this is potentially
indicative of a highly-obscured source. With conservative assumptions, and
accounting for both background fluctuations and the extended wings of the
quasar's emission, these photons only have a probability $P=0.021$ of happening
by chance. If confirmed by deeper observations, this system is the first high
redshift quasar and companion individually detected in X-rays and is likely a
dual AGN.
