  Using the large multi-wavelength data set in the chandra/SWIRE Survey (0.6
square degrees in the Lockman Hole), we show evidence for the existence of
highly obscured (Compton-thick) AGN, estimate a lower limit to their surface
density and characterize their multi-wavelength properties. Two independent
selection methods based on the X-ray and infrared spectral properties are
presented. The two selected samples contain 1) 5 X-ray sources with hard X-ray
spectra and column densities > 10^24 cm-2, and 2) 120 infrared sources with red
and AGN-dominated infrared spectral energy distributions (SEDs). We estimate a
surface density of at least 25 Compton-thick AGN per square degree detected in
the infrared in the chandra/SWIRE field of which ~40% show distinct AGN
signatures in their optical/near-infrared SEDs, the remainings being dominated
by the host-galaxy emission. Only ~33% of all Compton-thick AGN are detected in
the X-rays at our depth (F(0.3-8 keV)>10^-15 erg/cm2/s.
  We report the discovery of two sources in our sample of Compton-thick AGN,
SWIRE_J104409.95+585224.8 (z=2.54) and SWIRE_J104406.30+583954.1 (z=2.43),
which are the most luminous Compton-thick AGN at high-z currently known. The
properties of these two sources are discussed in detail with an analysis of
their spectra, SEDs, luminosities and black-hole masses.
