  We present Far Ultraviolet Spectroscopic Explorer and Hubble Space Telescope
observations of high, intermediate, and low ion absorption in high-velocity
cloud Complex C along the lines of sight toward five active galaxies. Our
purpose is to investigate the idea that Complex C is surrounded by an envelope
of highly ionized material, arising from the interaction between the cloud and
a hot surrounding medium. We measure column densities of high-velocity high ion
absorption, and compare the kinematics of low, intermediate, and high
ionization gas along the five sight lines. We find that in all five cases, the
H I and O VI high-velocity components are centered within 20 km/s of one
another, with an average displacement of <v(O VI)-v(H I)>=3+-12 km/s. In those
directions where the H I emission extends to more negative velocities (the
so-called high-velocity ridge), so does the O VI absorption. We compare our
high ion column density ratios to the predictions of various models, adjusted
to account for both recent updates to the solar elemental abundances, and for
the relative elemental abundance ratios in Complex C. Our observed ratios are
inconsistent with collisional ionization equilibrium at one kinetic
temperature. Photoionization by the extragalactic background is ruled out as
the source of the high ions, since the path lengths required would make Complex
C unreasonably large. We consider the most likely origin for the highly ionized
high-velocity gas to be at the conductive or turbulent interfaces between the
neutral/warm ionized components of Complex C and a surrounding hot medium. The
presence of interfaces on the surface of HVCs provides indirect evidence for
the existence of a hot medium in which the HVCs are immersed.
