  We present the first VLT/MUSE observations of a quasar featuring a proximate
molecular absorption system, SDSS J125917.31+030922.5. The proximate damped
Ly$\alpha$ absorption acts as a natural coronagraph, removing the quasar
emission over $\sim$40 {\AA} in wavelength, and allows us to detect extended
Ly$\alpha$ emission without the necessity of subtracting the quasar emission.
This natural coronagraph permits the investigation of the quasar environment
down to its inner regions ($r < 20$ kpc), where galaxy interactions or feedback
processes should have the most noticeable effects. Our observations reveal a
dense environment, with a highly asymmetric Ly$\alpha$ emission within $2"$
($\sim 15$ kpc), possibly shaped by a companion galaxy, and a southern
extension of the nebulae at about 50~kpc, with rotation-like kinematic
signature. The width of the Ly$\alpha$ emission is broadest closer to the
quasar, indicating perturbed kinematics as expected if interactions and
significant gas flows are present. The foreground absorbing system itself is
redshifted by $\approx $400 km/s relative to the background quasar, and
therefore is likely arising from gas moving towards the quasar. Finally two
additional Ly$\alpha$ emitters are detected with $>10\,\sigma$ significance at
96 and 223 kpc from the quasar, making this field overdense relative to other
similar observations of quasars at $z\sim 3$. Our results support the
hypothesis that quasars with proximate neutral/molecular absorption trace rich
environments where galaxy interactions are at play and motivates further
studies of H$_2$-selected quasars to shed light on feeding and feedback
processes.
