In this paper, we mainly study the high-order compact alternating direction implicit (ADI) scheme of the two-dimensional nonlinear Schrödinger equation with the Caputo fractional derivative. We adopt the L2-1 \(_{\sigma }\) formula to approximate the Caputo fractional derivative for the temporal discretization and the fourth-order compact finite difference method to approximate the second order spatial derivatives for the spatial discretization. In order to reduce the computational cost and memory, the nonlinear term is handled by local extrapolation method. By adding the splitting term, a compact ADI scheme is proposed for the model. The convergence of numerical schemes has been rigorously proven. Finally, a series of numerical experiments are proposed to confirm the theory results and simulate the dynamics of solution. It is not difficult to find that the proposed scheme converges with accuracy \(O(\tau ^{1+\alpha }+h_x^4+h_y^4)\) which is consistent with the theoretical result.
