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Denote by $\mu_\beta="\exp(\beta X)"$ the Gaussian multiplicative chaos whichis defined using a log-correlated Gaussian field $X$ on a domain$U\subset\mathbb{R}^d$. The case $\beta\in\mathbb{R}$ has been studied quiteintensively, and then $\mu_\beta$ is a random measure on $U$. It is known that$\mu_\beta$ can also be defined for complex values $\beta$ lying in certainsubdomain of $\mathbb{C}$, and then the realizations of $\mu_\beta$ are randomgeneralized functions on $U$. In this note we complement the results of Junnilaet al. (where the case of purely imaginary $\beta$ was considered) by studyingthe Besov-regularity of $\mu_\beta$ and the finiteness of moments for generalcomplex values of $\beta$.

On the regularity of complex multiplicative chaos