In this talk, I will discuss a newly proposed (pseudo-)critical phenomena governed by complex fixed points. I will start with the idea of complex fixed point at complex physical couplings and then introduce the recent conjectured complex conformal field theory with complex conformal data (e.g. central charge and scaling dimensions) which is suggested to describe these complex fixed points. These new concepts are putatively related to many interesting topics, such as the deconfined criticality, walking behavior in the gauge theories, weakly first order phase transitions and so on. Particularly, I will focus on a concrete example of the weakly first order phase transition, the Q>4 Potts model, where we did our numerics and found approximate conformality at intermediate length scale. Our results also give supportive evidence for the complex conformal data of recent conjectured complex CFTs.