Claim-augmented argumentation frameworks (CAFs) provide a formal basis to analyze conclusion-oriented problems in argumentation by adapting a claim-focused perspective; they extend Dung AFs by associating a claim to each argument representing its conclusion. This additional layer offers various possibilities to generalize abstract argumentation semantics as the re-interpretation of arguments in terms of their claims can be performed at different stages in the evaluation of the framework: One approach is to perform the evaluation entirely at argument-level before interpreting arguments by their claims (inherited semantics); alternatively, one can perform certain steps in the process (e.g., maximization) already in terms of the arguments’ claims (claim-level semantics). The inherent difference of these approaches not only potentially results in different outcomes but, as we will show in this paper, is also mirrored in terms of computational complexity. To this end, we provide a comprehensive complexity analysis of the four main reasoning problems with respect to claim-level variants of preferred, naive, stable, semi-stable and stage semantics and complete the complexity results of inherited semantics by providing corresponding results for semi-stable and stage semantics. Moreover, we show that deciding, whether for a given framework the two approaches of a semantics coincide (concurrence) can be surprisingly hard, ranging up to the third level of the polynomial hierarchy.