In this paper, we present a novel non-convex tensor completion model specifically tailored for multidimensional data. Our approach introduces a three-directional non-convex tensor rank surrogate regularized by the Minimax Concave Penalty (MCP) function. Crucially, the method processes data by simultaneously exploiting low-rank structures across its three modal directions, with the MCP function effectively mitigating the over-penalization of large singular values-a common drawback in convex nuclear norm minimization. To address the inherent challenges of this non-convex optimization, we develop an innovative approximate convex model that accurately captures the original formulation's essence. We then develop a robust convex Alternating Direction Method of Multipliers (ADMM)-based algorithm, supported by a rigorous convergence guarantee, ensuring both theoretical soundness and practical reliability. Extensive experiments on a variety of real-world datasets demonstrate the superior performance and robustness of the proposed method compared to state-of-the-art approaches.