  Gaussian Multiplicative Chaos (GMC) is informally defined as a random measure
$e^{\gamma X} \mathrm{d} x$ where $X$ is Gaussian field on $\mathbb R^d$ (or an
open subset of it) whose correlation function is of the form $ K(x,y)= \log
\frac{1}{|y-x|}+ L(x,y),$ where $L$ is a continuous function $x$ and $y$ and
$\gamma=\alpha+i\beta$ is a complex parameter. In the present paper, we
consider the case $\gamma\in \mathcal P'_{\mathrm{III}}$ where $$ \mathcal
P'_{\mathrm{III}}:= \{ \alpha+i \beta \ : \alpha,\gamma \in \mathbb R , \
|\alpha|<\sqrt{d/2}, \ \alpha^2+\beta^2\ge d \}.$$ We prove that if $X$ is
replaced by the approximation $X_\varepsilon$ obtained by convolution with a
smooth kernel, then $e^{\gamma X_\varepsilon} \mathrm d x$, when properly
rescaled, has an explicit non-trivial limit in distribution when $\varepsilon$
goes to zero. This limit does not depend on the specific convolution kernel
which is used to define $X_{\varepsilon}$ and can be described as a complex
Gaussian white noise with a random intensity given by a real GMC associated
with parameter $2\alpha$.
