We obtained ALMA spectroscopy and deep imaging to investigate the origin of the unexpected sub-millimeter emission toward the most distant quiescent galaxy known to date, ZF-COSMOS-20115 at
z= 3.717. We show here that this sub-millimeter emission is produced by another massive ( M*~ 1011 M), compact ( r12 = 0.67 0.14 kpc) and extremely obscured galaxy ( AV ~ 3.5), located only 0.43 (3.1 kpc) away from the quiescent galaxy. We dub the quiescent and dusty galaxies Jekyll and Hyde, respectively. No dust emission is detected at the location of the quiescent galaxy, implying SFR < 13 Myr1 which is the most stringent upper limit ever obtained for a quiescent galaxy at these redshifts. The two sources are spectroscopically confirmed to lie at the same redshift thanks to the detection of [C II]158 in Hyde ( z= 3.709), which provides one the few robust redshifts for a highly-obscured H-dropout galaxy ( H[4.5] = 5.1 0.8). The [C II] line shows a clear rotating-disk velocity profile which is blueshifted compared to the Balmer lines of Jekyll by 549 60 km s1, demonstrating that it is produced by another galaxy. Careful de-blending of the Spitzerimaging confirms the existence of this new massive galaxy, and its non-detection in the Hubbleimages requires extremely red colors and strong attenuation by dust. Full modeling of the UV-to-far-IR emission of both galaxies shows that Jekyll has fully quenched at least 200Myr prior to observation and still presents a challenge for models, while Hyde only harbors moderate star-formation with SFR 120 Myr1, and is located at least a factor 1.4 below the z~ 4 main sequence. Hyde could also have stopped forming stars less than 200 Myr before being observed; this interpretation is also suggested by its compactness comparable to that of z~ 4 quiescent galaxies and its low [C II]/FIR ratio, but significant on-going star-formation cannot be ruled out. Lastly, we find that despite its moderate SFR, Hyde hosts a dense reservoir of gas comparable to that of the most extreme starbursts. This suggests that whatever mechanism has stopped or reduced its star-formation must have done so without expelling the gas outside of the galaxy. Because of their surprisingly similar mass, compactness, environment and star-formation history, we argue that Jekyll and Hyde can be seen as two stages of the same quenching process, and provide a unique laboratory to study this poorly understood phenomenon.