  We present the ALMA detection of molecular outflowing gas in the central
regions of NGC4945, one of the nearest starbursts and also one of the nearest
hosts of an active galactic nucleus (AGN). We detect four outflow plumes in CO
(3-2) at ~0.3" resolution that appear to correspond to molecular gas located
near the edges of the known ionized outflow cone and its (unobserved)
counterpart behind the disk. The fastest and brightest of these plumes has
emission reaching observed line-of-sight projected velocities of over 450 km/s
beyond systemic, equivalent to an estimated physical outflow velocity v>600
km/s for the fastest emission. Most of these plumes have corresponding emission
in HCN or HCO+ (4-3). We discuss a kinematic model for the outflow emission
where the molecular gas has the geometry of the ionized gas cone and shares the
rotation velocity of the galaxy when ejected. We use this model to explain the
velocities we observe, constrain the physical speed of the ejected material,
and account for the fraction of outflowing gas that is not detected due to
confusion with the galaxy disk. We estimate a total molecular mass outflow rate
dMmol/dt~20 Msun/yr flowing through a surface within 100 pc of the disk
midplane, likely driven by a combination of the central starburst and AGN.
