  In this paper, we study complex valued branching Brownian motion in the
so-called glassy phase, or also called phase II. In this context, we prove a
limit theorem for the complex partition function hence confirming a conjecture
formulated by Lacoin and the last two authors in a previous paper on complex
Gaussian multiplicative chaos. We will show that the limiting partition
function can be expressed as a product of a Gaussian random variable, mainly
due to the windings of the phase, and a stable transform of the so called
derivative martingale, mainly due to the clustering of the modulus. The proof
relies on the fine description of the extremal process available in the
branching Brownian motion context.
