This paper introduces discrete-holomorphic Perfectly Matched Layers (PMLs) specifically designed for high-order finite difference (FD) discretizations of the scalar wave equation. In contrast to standard PDE-based PMLs, the proposed method achieves the remarkable outcome of completely eliminating numerical reflections at the PML interface, in practice achieving errors at the level of machine precision. Our approach builds upon the ideas put forth in a recent publication [Journal of Computational Physics 381 (2019): 91-109] expanding the scope from the standard second-order FD method to arbitrary high-order schemes. This generalization uses additional localized PML variables to accommodate the larger stencils employed. We establish that the numerical solutions generated by our proposed schemes exhibit an exponential decay rate as they propagate within the PML domain. To showcase the effectiveness of our method, we present a variety of numerical examples, including waveguide problems. These examples highlight the importance of employing high-order schemes to effectively address and minimize undesired numerical dispersion errors, emphasizing the practical advantages and applicability of our approach.

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