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This paper presents a fully complex-valued functional link network (CFLN). The CFLN is a single-layered neural network, which introduces nonlinearity in the input layer using nonlinear functions of the original input variables. In this study, we consider multivariate polynomials as the nonlinear functions. Unlike multilayer neural networks, the CFLN is free from local minima problem, and it offers very fast learning in parameters because of its linear structure. In the complex domain, polynomial based CFLN has an additional advantage of not requiring activation functions, which is a major concern in the complex-valued neural networks. However, it is important to select a smaller subset of polynomial terms (monomials) for faster and better performance, since the number of all possible monomials may be quite large. In this paper, we use the orthogonal least squares method in a constructive fashion (starting from lower degree to higher) for the selection of a parsimonious subset of monomials. Simulation results demonstrate that computing CFLN in purely complex domain is advantageous than in double-dimensional real domain, in terms of number of connection parameters, faster design, and possibly generalization performance. Moreover, our proposed CFLN compares favorably with several other multilayer networks in the complex domain.