  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.
