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We obtained ALMA spectroscopy and imaging to investigate the origin of theunexpected sub-mm emission toward the most distant quiescent galaxy known todate, ZF-COSMOS-20115 at z=3.717. We show here that this sub-mm emission isproduced by another massive, compact and extremely obscured galaxy, locatedonly 3.1 kpc away from the quiescent galaxy. We dub the quiescent and dustygalaxies Jekyll and Hyde, respectively. No dust emission is detected at thelocation of Jekyll, implying SFR < 13 Msun/yr, which is the most stringentupper limit ever obtained for a quiescent galaxy at these redshifts. The twosources are confirmed to lie at the same redshift thanks to the detection of[CII]158 in Hyde, which provides one the few robust redshifts for an"H-dropout" galaxy. The line has a rotating-disk velocity profile blueshiftedfrom Jekyll by 549+/-60 km/s, demonstrating that it is produced by anothergalaxy. Careful de-blending of the Spitzer imaging confirms the existence ofHyde, and its non-detection with Hubble requires extreme attenuation by dust.Modeling the photometry of both galaxies shows that Jekyll has fully quenched>200 Myr prior to observation and still presents a challenge for models, whileHyde only harbors moderate star-formation (SFR<120 Msun/yr) and is located atleast a factor 1.4 below the z~4 main sequence. Hyde could also have stoppedforming stars <200 Myr before being observed, which would be consistent withits hight compactness similar to z~4 quiescent galaxies and its low [CII]/FIRratio, but significant SF cannot be ruled out. Finally, we show that Hyde hostsa dense reservoir of gas comparable to that of extreme starbursts, suggestingthat its SFR was reduced without expelling the gas outside of the galaxy. Weargue that Jekyll and Hyde can be seen as two stages of quenching, and providea unique laboratory to study this poorly understood phenomenon. [abridged]

Jekyll & Hyde: quiescence and extreme obscuration in a pair of massive galaxies 1.5 Gyr after the Big Bang