We report the discovery of significant linear polarization (p > 1%) in the hyperluminous z = 3.87 broad absorption line (BAL) QSO 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 Angstrom (rest) supports the interpretation of this feature as a broad absorption component associated with O VI/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 sight line 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.