The Second Order Kalman Filter (SOKF) uses a second order Taylor series expansion (TSE) to account for nonlinearities in an estimation problem. In this work, the derivatives required for the SOKF are computed using multicomplex (MCX) derivatives, coded in the Matlab programming language. This method uses function overloading in order to derive or compute the derivatives to machine precision without having to compute the derivatives analytically. Thus, the SOKF can be easily implemented, while at the same time having fewer tuning parameters than other high order filters. The standard SOKF is also extended by combining it with Gaussian Mixture models (GMM), which gives promising results. The filters have been used to estimate the state of a 1 DOF falling body. The results show that the MCX computes the required derivatives just as accurately as an analytical method and the SOKF and GMM modification perform well in terms of accuracy compared to other filters. Despite the ease of use and high accuracy benefits, a current drawback of the MCX method is compute speed. Methods for improving the speed are beyond the current scope and will be addressed in future works.