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)).
