  We consider the imaginary Gaussian multiplicative chaos, i.e. the complex
Wick exponential $\mu_\beta := :e^{i\beta \Gamma(x)}:$ for a log-correlated
Gaussian field $\Gamma$ in $d \geq 1$ dimensions. We prove a basic density
result, showing that for any nonzero continuous test function $f$, the
complex-valued random variable $\mu_\beta(f)$ has a smooth density w.r.t. the
Lebesgue measure on $\mathbb{C}$. As a corollary, we deduce that the negative
moments of imaginary chaos on the unit circle do not correspond to the analytic
continuation of the Fyodorov-Bouchaud formula, even when well-defined.
  Somewhat surprisingly, basic density results are not easy to prove for
imaginary chaos and one of the main contributions of the article is introducing
Malliavin calculus to the study of (complex) multiplicative chaos. To apply
Malliavin calculus to imaginary chaos, we develop a new decomposition theorem
for non-degenerate log-correlated fields via a small detour to operator theory,
and obtain small ball probabilities for Sobolev norms of imaginary chaos.
