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Kernel-based adaptive filters present a new opportunity to re-cast nonlinear optimization problems over a Reproducing Kernel Hilbert Space (RKHS), transforming the nonlinear task to linear, where easier and well-known methods may be used. The approach can be seen to yield solutions suitable for sparse adaptive filtering. The new Complex Kernel Least Mean Square algorithm (CKLMS), derived by Bouboulis and Theodoridis, allows kernel-based online adaptive filtering for complex data. Here we report our results on the convergence of CKLMS with the complexified form of the Gaussian kernel. The analysis performed is based on a recent study of the Kernel LMS from Parreira et al. The analysis is used to generate theory-predicted MSE curves which consider the circularity/non-circularity of complex input which to our knowledge has not been considered previously for online nonlinear learning. Simulations are used to verify the theoretical analysis results.