In this paper, we investigate the coupled modified nonlinear Schödinger equation. Through the traditional Darboux transformation, we construct the first-order breather solution which can exhibit Akhmediev breather and general breather. To obtain the higher-order localized wave solution, N-fold generalized Darboux transformation is given. Under the condition that the characteristic equation admits a double-root, we present the expression of first-order interactional solution. Then we graphically analyze the dynamics of breather and rogue wave. Due to the simultaneous existence of nonlinear and derivative terms in the equation, there presents different profile in two components for the breathers.