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In this paper, we use a biquaternion formalism to model vector-sensor signals carrying polarization information. This allows a concise and elegant way of handling signals with eight-dimensional (8-D) vector-valued samples. Using this model, we derive a biquaternionic version of the well-known array processing MUSIC algorithm, and we show its superiority to classically used long-vector approach. New results on biquaternion valued matrix spectral analysis are presented. Of particular interest for the biquaternion MUSIC (BQ-MUSIC) algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces. We propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition (EVD) of a Hermitian biquaternionic matrix by means of a newly defined quantity, the quaternion adjoint matrix. The BQ-MUSIC estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing (LV-MUSIC and PSA-MUSIC). The proposed algorithm is shown to be superior in several aspects to the existing approaches. Compared with LV-MUSIC, the BQ-MUSIC algorithm is more robust to modelization errors and coherent noise while it can detect less sources. In comparaison with PSA-MUSIC, our approach exhibits more accurate estimation of direction of arrival (DOA) for a small number of sources, while keeping the polarization information accessible.