  Conformal symmetry, emerging at critical points, can be lost when
renormalization group fixed points collide. Recently, it was proposed that
after collisions, real fixed points transition into the complex plane, becoming
complex fixed points described by complex conformal field theories (CFTs).
Although this idea is compelling, directly demonstrating such complex conformal
fixed points in microscopic models remains highly demanding. Furthermore, these
concrete models are instrumental in unraveling the mysteries of complex CFTs
and illuminating a variety of intriguing physical problems, including weakly
first-order transitions in statistical mechanics and the conformal window of
gauge theories. In this work, we have successfully addressed this complex
challenge for the (1+1)-dimensional quantum $5$-state Potts model, whose phase
transition has long been known to be weakly first-order. By adding an
additional non-Hermitian interaction, we successfully identify two conjugate
critical points located in the complex parameter space, where the lost
conformality is restored in a complex manner. Specifically, we unambiguously
demonstrate the radial quantization of the complex CFTs and compute the
operator spectrum, as well as new operator product expansion coefficients that
were previously unknown.
