  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$_{\rm bol}$ $\ge$ 10$^{14}$ L$_\odot$)
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 $\Sigma_{\rm [CII]}$
~ 1-2 x 10$^{9}$ L$_\odot$ 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 $\pm$ 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 $\gtrsim$ 500 km/s,
which on average are larger than optically and IR-selected quasars at similar
or higher redshifts. We argue that one 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
single event, the EL Hot DOG phase could be recurrent.
