In this paper, the analytical solutions of the complex coupled Higgs field equation, which explains a system of conserved scalar nucleons that interact with neutral scalar mesons in particle physics, are extracted. The enhanced modified extended tanh expansion method (eMETEM) is successfully employed to solve the considered equation. Besides, the effect of problem parameters on a real scalar meson field and a complex scalar nucleon field is investigated, plotted and interpreted. Some effective results are obtained in the examination and are presented for the first time in this study. The obtained results can be applied to a wide range of nonlinear equations including ocean and shallow-water waves. 3D, 2D and contour graphical representations of the attained solutions are given using appropriate parameters. The method is efficient for gathering analytical solutions for various kinds of phenomena modeling by classical, fractional, or stochastic partial differential equations.