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We have obtained >10 hr of medium-resolution (R ∼ 15,000) spectroscopic exposures on the transiting exoplanet host star WASP-12, including ∼2 hr while its planet, WASP-12b, is in transit, with the Hobby-Eberly Telescope. The out-of-transit and in-transit spectra are coadded into master out-of-transit and in-transit spectra, from which we create a master transmission spectrum. Strong, statistically significant absorption features are seen in the transmission spectrum at Hα and Na I (the Na D doublet). There is the suggestion of pre- and post-transit absorption in both Hα and Na I when the transmission spectrum is examined as a function of phase. The timing of the pretransit absorption is roughly consistent with previous results for metal absorption in WASP-12b, and the level of the Na I absorption is consistent with a previous tentative detection. No absorption is seen in the control line of Ca I at λ6122. We discuss in particular whether or not the WASP-12b Hα absorption signal is of circumplanetary origin-an interpretation that is bolstered by the pre- and post-transit evidence-which would make it one of only a small number of detections of circumplanetary Hα absorption in an exoplanet to date, the most well-studied being HD 189733b. We further discuss the notable differences between the HD 189733 and WASP-12 systems and the implications for a physical understanding of the origin of the absorption.