Quaternion algebra presents a promising mathematical modeling and computation framework, offering a natural extension to real and complex numbers. This paper introduces a free module structure to represent quaternions in complex settings. This new structure is used to extend the conventional subspace iteration method to quaternion matrix spaces. Thus, a low-rank subspace iteration method is proposed to solve iteratively a quaternion least squares minimization problem. Finally, we apply our proposed method to the task of color image completion, showcasing its practical utility.