The cluster editing problem is a problem of converting a given undirected graph into a set of cliques. The chromatic correlation clustering (CCC) problem is a generalization of the cluster editing problem in which the edges are assigned colors and the aim is to create monochromatic cliques. The problem is NP-hard, but a number of approximation algorithms for the problem are developed. In contrast, no fixed-parameter tractable (FPT) algorithm for the CCC problem has yet been developed. We present a novel framework that extends existing branching algorithms for cluster editing to solve chromatic correlation clustering. We ensure that the extension causes no degradation in the overall time complexity. To demonstrate its effectiveness, we apply the framework to an existing algorithm, thus creating an FPT algorithm for CCC with the same time complexity.