Multi-look acquisition is a widely used strategy for reducing speckle noise in coherent imaging systems such as digital holography. By acquiring multiple measurements, speckle can be suppressed through averaging or joint reconstruction, typically under the assumption that speckle realizations across looks are statistically independent. In practice, however, hardware constraints limit measurement diversity, leading to inter-look correlation that degrades the performance of conventional methods. In this work, we study the reconstruction of speckle-free reflectivity from complex-valued multi-look measurements in the presence of correlated speckle. We model the inter-look dependence using a first-order Markov process and derive the corresponding likelihood under a first-order Markov approximation, resulting in a constrained maximum likelihood estimation problem. To solve this problem, we develop an efficient projected gradient descent framework that combines gradient-based updates with implicit regularization via deep image priors, and leverages Monte Carlo approximation and matrix-free operators for scalable computation. Simulation results demonstrate that the proposed approach remains robust under strong inter-look correlation, achieving performance close to the ideal independent-look scenario and consistently outperforming methods that ignore such dependencies. These results highlight the importance of explicitly modeling inter-look correlation and provide a practical framework for multi-look holographic reconstruction under realistic acquisition conditions. Our code is available at: https://github.com/Computational-Imaging-RU/MLE-Holography-Markov.