In this paper, cluster synchronization in finite/fixed time for semi-Markovian switching complex dynamical networks (CDNs) with discontinuous dynamic nodes is studied. Firstly, the global fixed-time convergence principle of nonlinear systems with semi-Markovian switching is developed. Secondly, the novel state-feedback controllers, which include discontinuous factors and integral terms, are designed to achieve the global stochastic finite/fixed-time cluster synchronization. In the framework of Filippov stochastic differential inclusion, the Lyapunov-Krasovskii functional approach, Takagi-Sugeno(T-S) fuzzy theory, stochastic analysis theory, and inequality analysis techniques are applied, and the global stochastic finite/fixed time synchronization conditions are proposed in the form of linear matrix inequalities (LMIs). Moreover, the upper bound of the stochastic settling time is explicitly proposed. In addition, the correlations among the obtained results are interpreted analytically. Finally, two numerical examples are given to illustrate the correctness of the theoretical results.