This paper addresses the challenge of computing right eigenvalues for non-Hermitian dual quaternion matrices (DQMs). The non-Hermitian case introduces significant complexity, as such a matrix may possess either no eigenvalues or infinitely many. Furthermore, the eigenvalues themselves are dual quaternion numbers, whose non-commutative nature further complicates the analysis. We extend the power method from the Hermitian DQMs [10] to the non-Hermitian setting. First, we establish a sufficient condition that guarantees the existence of an eigenvalue. Under a stronger, more restrictive condition, we prove that the sequence generated by the power method converges linearly to the strict dominant eigenvalue. Notably, we also demonstrate that this condition is both necessary and sufficient. The key to our analysis is a novel Jordan-like decomposition, which addresses a gap arising from the lack of a conventional Jordan decomposition for non-Hermitian dual matrices. Our framework readily extends to non-Hermitian dual complex and dual number matrices. We also develop an adjoint method that reformulates the eigenvalue problem into an equivalent form of dual complex matrices. Numerical experiments are presented to demonstrate the efficiency of the power method. Our code is available at https://github.com/BUAA-HaoYang/DQ-toolbox
