Strong optical anisotropy is observed in the emission from a GaAs1−xBix (x ∼ 0.04) quantum well grown by low temperature molecular beam epitaxy on (001) GaAs by means of low temperature magneto-photoluminescence (MPL) taken at 2 K in Faraday geometry for magnetic fields, B, up to 10 T. A significant diamagnetic shift (∼2.5 meV) develops for magnetic fields above ∼8 T, which is accompanied by a narrowing of the emission bandwidth and a substantial increase in the difference between the integrated intensities of the σ+ and σ− polarizations in the MPL spectra. This, along with a peculiar spectral dependence of the polarization degree which evolves with increasing magnetic field, is interpreted in terms of bound and free magneto excitons in the system where Bi-related levels become hybridized to different extents with the valence and conduction bands of the GaAs host material.