The quasar SDSS J105041.35+345631.3 (z = 0.272) has broad emission lines blueshifted by 3500 km/s relative to the narrow lines and the host galaxy. Such an object may be a candidate for a recoiling supermassive black hole, binary black hole, a superposition of two objects, or an unusual geometry for the broad emission-line region. The absence of narrow lines at the broad line redshift argues against superposition. New Keck spectra of J1050+3546 place tight constraints on the binary model. The combination of large velocity shift and symmetrical H-beta profile, as well as aspects of the narrow line spectrum, make J1050+3546 an interesting candidate for black hole recoil. Other aspects of the spectrum, however, suggest that the object is most likely an extreme case of a ``double-peaked emitter. We discuss possible observational tests to determine the true nature of this exceptional object.
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