  In CP-violating conformal field theories in four dimensions, the Pontryagin
density can appear in the Weyl anomaly. The Pontryagin density in the Weyl
anomaly is consistent, but it has a peculiar feature that the parent
three-point function of the energy-momentum tensor can violate CP only
(semi-)locally. In this paper, we study the supersymmetric completion of the
Pontryagin density in the Weyl anomaly, where the central charge $c$
effectively becomes a complex number. The supersymmetry suggests that it
accompanies the graviphoton $\theta$ term associated with the R-symmetry
gauging in the Weyl anomaly. It also accompanies new CP-violating terms in the
R-current anomaly. While there are no conclusive perturbative examples of
CP-violating super Weyl anomaly, we construct explicit supersymmetric dilaton
effective action which generates these anomalies.
