This paper shows the problem of model of coherence in the quaternion model when using quaternion multiple signal classification algorithm (Q-MUSIC) for the direction of arrival (DOA) of polarized signals estimation with uniform linear array (ULA), which consists of dual dipole antennas, and solve the above problem effectively by proposing the real isomorphic matrix transformation MUSIC algorithm with stretched dual dipole antennas. The proposed algorithm makes full use of the orthogonality of the quaternion model and has a superior performance to the existing Q-MUSIC algorithm based on the conventional dual dipole antenna especially for low signal to noise ratio (SNR). Finally, the numerical simulations show the effectiveness of the proposed method.