  Denote by $\mu_\beta="\exp(\beta X)"$ the Gaussian multiplicative chaos which
is 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 quite
intensively, 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 certain
subdomain of $\mathbb{C}$, and then the realizations of $\mu_\beta$ are random
generalized functions on $U$. In this note we complement the results of Junnila
et al. (where the case of purely imaginary $\beta$ was considered) by studying
the Besov-regularity of $\mu_\beta$ and the finiteness of moments for general
complex values of $\beta$.
