Under investigation in this work is the general coupled nonlinear Schr?dinger(g CNLS) equation, which can be used to describe a wide variety of physical processes. By using Darboux transformation, the new higher-order rogue wave solutions of the equation are well constructed. These solutions exhibit rogue waves on a multi-soliton background.Moreover, the dynamics of these solutions is graphically discussed. Our results would be of much importance in enriching and predicting rogue wave phenomena arising in nonlinear and complex systems.