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.
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