Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemannian manifolds. We discuss one of the important geometric observables: critical points. We compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius. We observe a similar behaviour for the point process of critical points of isotropic stationary Gaussian fields.