Personalized federated learning (pFL) is to collaboratively train non-identical machine learning models for different clients to adapt to their heterogeneously distributed datasets. State-of-the-art pFL approaches pay much attention on exploiting clients’ inter-similarities to facilitate the collaborative learning process, meanwhile, can barely escape from the irrelevant knowledge pooling that is inevitable during the aggregation phase, and thus hindering the optimization convergence and degrading the personalization performance. To tackle such conflicts between facilitating collaboration and promoting personalization, we propose a novel pFL framework, dubbed pFedC, to first decompose the global aggregated knowledge into several compositional branches, and then selectively reassemble the relevant branches for supporting conflicts-aware collaboration among contradictory clients. Specifically, by reconstructing each local model into a shared feature extractor and multiple decomposed task-specific classifiers, the training on each client transforms into a mutually reinforced and relatively independent multi-task learning process, which provides a new perspective for pFL. Besides, we conduct a purified knowledge aggregation mechanism via quantifying the combination weights for each client to capture clients’ common prior, as well as mitigate potential conflicts from the divergent knowledge caused by the heterogeneous data. Extensive experiments over various models and datasets demonstrate the effectiveness and superior performance of the proposed algorithm.