We study complex CFTs describing fixed points of the two-dimensional Q-state Potts model with Q > 4. Their existence is closely related to the weak first-order phase transition and the "walking" renormalization group (RG) behavior present in the real Potts model at Q > 4. The Potts model, apart from its own significance, serves as an ideal playground for testing this very general relation. Cluster formulation provides nonperturbative definition for a continuous range of parameter Q, while Coulomb gas description and connection to minimal models provide some conformal data of the complex CFTs. We use one and two-loop conformal perturbation theory around complex CFTs to compute various properties of the real walking RG flow. These properties, such as drifting scaling dimensions, appear to be common features of the QFTs with walking RG flows, and can serve as a smoking gun for detecting walking in Monte Carlo simulations. The complex CFTs discussed in this work are perfectly well defined, and can in principle be seen in Monte Carlo simulations with complexified coupling constants. In particular, we predict a pair of S-5-symmetric complex CFTs with central charges c approximate to 1.138 +/- 0.021i describing the fixed points of a 5-state dilute Potts model with complexified temperature and vacancy fugacity. Copyright V. Gorbenko et al.