  We report the discovery of significant linear polarization (p > 1%) in the
hyperluminous z=3.87 BALQSO APM~08279+5255. The polarization spectrum is
complex, with properties similar to those of other, lower redshift but more
highly polarized BALQSOs. The resonance emission lines are unpolarized while
the absorption troughs show polarization similar to or higher than the
continuum. In particular, an apparent increase of polarization in the trough
covering 1000-1030 angstroms (rest) supports the interpretation of this feature
as a broad absorption component associated with OVI/Ly_beta local to the QSO,
as opposed to an intervening damped Ly_alpha absorption system.
  The elevated polarization in some of the absorption features implies that we
view the scattered (polarized) spectrum through a sightline with less absorbing
material than the direct spectrum. Therefore, the complex structure of the
polarization spectrum in this brilliant lensed BALQSO suggests that it will be
an important laboratory for studying the structure of QSOs at high redshift.
