This paper focuses on the synchronization issues of stochastic multi-weights complex networks associated with Lévy noise and Markov chains (SMWCNLM). Firstly, a synchronization criterion is obtained for the complex networks without the noisy time delay using the Lyapunov stability method and graph theory. Secondly, a sufficient condition is established to guarantee almost sure synchronization of the complex networks with the noisy time delay based on the non-negative semi-martingales convergence theorem in the framework of the stochastic process. Meanwhile, the coupling structure of all systems is controlled by continuous-time homogeneous Markov chains. To realize the synchronization of the systems, state feedback controllers are employed and these obtained synchronization principles are correlated to the multi-weights and the intensity of the noise. Finally, numerical simulations are presented to validate these theoretical results.