In this paper, we analyze the finite-time synchronization problem between two delayed diffusive complex-valued neural networks(CVNNs) with discontinuous activations. We first establish the threshold finite-/fixed-time synchronization(FFTS) dynamics of the model by designing a novel negative exponent controller. Then we further study the finite-time synchronization via the adaptive control scheme. Some novel and useful finite-time synchronization criteria are established based on the discontinuous version of finite-time convergence theorem and Filippov regularization techniques, the upper-bound...