This paper deals with the numerical solutions of two-dimensional (2D) semi-linear reaction-diffusion equations (SLRDEs) with piecewise continuous argument (PCA) in reaction term. A high-order compact difference method called I-type basic scheme is developed for solving the equations and it is proved under the suitable conditions that this method has the computational accuracy $\mathcal{O}(\tau^2+h^4_x+h^4_y),$ where $\tau,$ $h_x$ and $h_y$ are the calculation stepsizes of the method in $t$-, $x$- and $y$-direction, respectively. With the above method and Newton linearized technique, a ${\rm II}$-type basic scheme is also suggested. Based on the both basic schemes, the corresponding ${\rm I}$- and ${\rm II}$-type alternating direction implicit (ADI) schemes are derived. Finally, with a series of numerical experiments, the computational accuracy and efficiency of the four numerical schemes are further illustrated.