  Hot, dust-obscured galaxies (Hot DOGs) are a population of hyper-luminous
obscured quasars identified by WISE. We present ALMA observations of the [CII]
fine-structure line and underlying dust continuum emission in a sample of seven
of the most extremely luminous (EL; L_bol >= 10^14 L_sun) Hot DOGs, at
redshifts z ~ 3.0-4.6. The [CII] line is robustly detected in four objects,
tentatively in one, and likely red-shifted out of the spectral window in the
remaining two based on additional data. On average, [CII] is red-shifted by ~
780 km/s from rest-frame ultraviolet emission lines. EL Hot DOGs exhibit
consistently very high ionized gas surface densities, with Sigma_[CII] ~ 1-2 x
10^9 L_sun/kpc^2; as high as the most extreme cases seen in other high-redshift
quasars. As a population, EL Hot DOG hosts seem to be roughly centered on the
main-sequence of star forming galaxies, but the uncertainties are substantial
and individual sources can fall above and below. The average, intrinsic [CII]
and dust continuum sizes (FWHMs) are ~ 2.1 kpc and ~ 1.6 kpc, respectively,
with a very narrow range of line-to-continuum size ratios, 1.61 +/- 0.10,
suggesting they could be linearly proportional. The [CII] velocity fields of EL
Hot DOGs are diverse: from barely rotating structures, to resolved hosts with
ordered, circular motions, to complex, disturbed systems that are likely the
result of ongoing mergers. In contrast, all sources display large line-velocity
dispersions, FWHM >~ 500 km/s, which on average are larger than optically and
IR-selected quasars at similar or higher redshifts. We argue that a possible
hypothesis for the lack of a common velocity structure, the systematically
large dispersion of the ionized gas, and the presence of nearby companion
galaxies may be that, rather than a continuous single event, the EL Hot DOG
phase could be recurrent.
