  To understand the Active Galactic Nuclei (AGN) phenomenon and their impact on
the evolution of galaxies, a complete AGN census is required; however, finding
heavily obscured AGNs is observationally challenging. Here we use the deep and
extensive multi-wavelength data in the COSMOS field to select a complete sample
of 578 infrared (IR) quasars ($L_{\rm AGN,IR}>10^{45}\rm \: erg\: s^{-1}$) at
$z<3$, with minimal obscuration bias, using detailed UV-to-far IR spectral
energy distribution (SED) fitting. We complement our SED constraints with X-ray
and radio observations to further investigate the properties of the sample.
Overall, 322 of the IR quasars are detected by Chandra and have individual
X-ray spectral constraints. From a combination of X-ray stacking and $L_{\rm
2-10\rm keV}$ - $L_{\rm 6\: \mu m}$ analyses, we show that the majority of the
X-ray faint and undetected quasars are heavily obscured (many are likely
Compton thick), highlighting the effectiveness of the mid-IR band to find
obscured AGNs. We find that 355 ($\approx$61%) IR quasars are obscured ($N_{\rm
H}>10^{22}\rm \: cm^{-2}$) and identify differences in the average properties
between the obscured and unobscured quasars: (1) obscured quasars have
star-formation rates $\approx 3$ times higher than unobscured systems for no
significant difference in stellar mass and (2) obscured quasars have stronger
radio emission than unobscured systems, with a radio-loudness parameter
$\approx 0.2 \rm \: dex$ higher. These results are inconsistent with a simple
orientation model but in general agreement with either extreme host-galaxy
obscuration towards the obscured quasars or a scenario where obscured quasars
are an early phase in the evolution of quasars.
