  This paper focuses on the modulation instability, conservation laws and
localized wave solutions of the generalized coupled Fokas-Lenells equation.
Based on the theory of linear stability analysis, distribution pattern of
modulation instability gain G in the (K,k) frequency plane is depicted, and the
constraints for the existence of rogue waves are derived. Subsequently, we
construct the infinitely many conservation laws for the generalized coupled
Fokas-Lenells equation from the Riccati-type formulas of the Lax pair. In
addition, the compact determinant expressions of the N-order localized wave
solutions are given via generalized Darboux transformation, including
higher-order rogue waves and interaction solutions among rogue waves with
bright-dark solitons or breathers. These solutions are parameter controllable:
(m_i,n_i) and (\alpha,\beta) control the structure and ridge deflection of
solution respectively, while the value of |d| controls the strength of
interaction to realize energy exchange. Especially, when d=0, the interaction
solutions degenerate into the corresponding order of rogue waves.
