  We obtained ALMA spectroscopy and imaging to investigate the origin of the
unexpected sub-mm emission toward the most distant quiescent galaxy known to
date, ZF-COSMOS-20115 at z=3.717. We show here that this sub-mm emission is
produced by another massive, compact and extremely obscured galaxy, located
only 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 Jekyll, implying SFR < 13 Msun/yr, which is the most stringent
upper limit ever obtained for a quiescent galaxy at these redshifts. The two
sources 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 blueshifted
from Jekyll by 549+/-60 km/s, demonstrating that it is produced by another
galaxy. Careful de-blending of the Spitzer imaging confirms the existence of
Hyde, 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, while
Hyde only harbors moderate star-formation (SFR<120 Msun/yr) and is located at
least a factor 1.4 below the z~4 main sequence. Hyde could also have stopped
forming stars <200 Myr before being observed, which would be consistent with
its hight compactness similar to z~4 quiescent galaxies and its low [CII]/FIR
ratio, but significant SF cannot be ruled out. Finally, we show that Hyde hosts
a dense reservoir of gas comparable to that of extreme starbursts, suggesting
that its SFR was reduced without expelling the gas outside of the galaxy. We
argue that Jekyll and Hyde can be seen as two stages of quenching, and provide
a unique laboratory to study this poorly understood phenomenon. [abridged]
