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We present Far Ultraviolet Spectroscopic Explorer and Hubble Space Telescope observations of high-, intermediate-, and low-ion absorption in high-velocity cloud (HVC) 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{d1}_OVI_–v{d1}HI_=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. The kinematics of Si II is also similar to that of O VI, with v{d1}_OVI_–v{d1}SiII_=0±15 km/s. 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 relative elemental abundance ratios in Complex C. Along the PG 1259+593 sight line, we measure N(SiIV)/N(OVI)=0.10±0.02, N(CIV)/N(OVI)=0.35^+0.05_–0.06, and N(NV)/N(OVI)<0.07 (3 σ). These 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 HVCs unreasonably large; photoionization by radiation from the disk of the Galaxy also appears unlikely since the emerging photons are not energetic enough to produce O VI. By themselves, ionic ratios are insufficient to discriminate between various ionization models, but by considering the absorption kinematics as well, 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. This medium could be a hot (T≳10⁶ K) extended Galactic corona or hot gas in the Local Group.