It is known that all continuous orbital measures, $\mu$ on a compact, connected, classical simple Lie group $G$ or its Lie algebra satisfy a dichotomy: either $\mu^{k} \in L^{2}$ or $\mu^{k}$ is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group $G^C$. We also determine the sharp exponent $k$ such that any $k$-fold convolution product of continuous $G$-bi-invariant measures on $G^{C}$ is absolute continuous with respect to Haar measure.