<p>Biquaternion algebra is an extension of quaternion algebra. During recent years, it has attracted increasing attention in fields such as signal processing and computer vision. In this paper, we introduce a QR decomposition algorithm for biquaternion matrices for the first time, using a complex representation and an odd–even row/column separation strategy. Experiments demonstrate that our algorithm is both efficient and accurate for computing the QR decomposition of biquaternion matrices. We further apply it to quaternion matrices and compare it with state-of-the-art algorithms for computing the QR decomposition of quaternion matrices. The results show that the proposed method offers clear advantages in accuracy and computational efficiency, particularly in CPU time. As applications, we employ it for simultaneous encryption/ decryption of two images, as well as for watermark embedding/ extraction, which confirms its practical usefulness.</p>