  In this note we continue the study of imaginary multiplicative chaos
$\mu_\beta := \exp(i \beta \Gamma)$, where $\Gamma$ is a two-dimensional
continuum Gaussian free field. We concentrate here on the fine-scale analytic
properties of $|\mu_\beta(Q(x,r))|$ as $r \to 0$, where $Q(x,r)$ is a square of
side-length $2r$ centred at $x$. More precisely, we prove monofractality of
this process, a law of the iterated logarithm as $r \to 0$ and analyse its
exceptional points, which have a close connection to fast points of Brownian
motion. Some of the technical ideas developed to address these questions also
help us pin down the exact Besov regularity of imaginary chaos, a question left
open in [JSW20]. All the mentioned properties illustrate the noise-like
behaviour of the imaginary chaos. We conclude by proving that the processes $x
\mapsto |\mu_\beta(Q(x,r))|^2$, when normalised additively and
multiplicatively, converge as $r \to 0$ in law, but not in probability, to
white noise; this suggests that all the information of the multiplicative chaos
is contained in the angular parts of $\mu_\beta(Q(x,r))$.
