The complex Gaussian multiplicative chaos (or complex GMC) is informally defined as a random
distribution
${e}^{\gamma X}dx$
where
$X$ is a
$\mathrm{log}$correlated
Gaussian field on
${\mathbb{R}}^{d}$
and
$\gamma =\alpha +i\beta $
is a complex parameter. The correlation function of
$X$ is of
the form
$$K(x,y)=\mathrm{log}\frac{1}{xy}+L(x,y),$$ 
where
$L$
is a continuous function. We consider the cases
$\gamma \in {\mathcal{\mathcal{P}}}_{\mathrm{I/II}}^{}$ and
$\gamma \in {\mathcal{\mathcal{P}}}_{\mathrm{II/III}}^{\prime}$ where
$${\mathcal{\mathcal{P}}}_{\mathrm{I/II}}^{}:=\left\{\alpha +i\beta :\alpha ,\beta \in \mathbb{R};\left\alpha \right\in \left(\sqrt{d\u22152},\sqrt{2d}\right);\left\alpha \right+\left\beta \right=\sqrt{2d}\right\}$$ 
and
$${\mathcal{\mathcal{P}}}_{\mathrm{II/III}}^{\prime}:=\left\{\alpha +i\beta :\alpha ,\beta \in \mathbb{R};\left\alpha \right=\sqrt{d\u22152};\left\beta \right\ge \sqrt{d\u22152}\right\}.$$ 
We prove that if
$X$ is replaced
by an approximation
${X}_{\mathit{\epsilon}}$ obtained
via mollification, then
${e}^{\gamma {X}_{\mathit{\epsilon}}}dx$, when
properly rescaled, converges when
$\mathit{\epsilon}\to 0$.
The limit does not depend on the mollification kernel. When
$\gamma \in {\mathcal{\mathcal{P}}}_{\mathrm{I/II}}^{}$, the convergence holds
in probability and in
${L}^{p}$
for some value of
$p\in \left[1,\sqrt{2d}\u2215\alpha \right)$.
When
$\gamma \in {\mathcal{\mathcal{P}}}_{\mathrm{II/III}}^{\prime}$,
the convergence holds only in law. In this latter case, the limit can be described as a
complex Gaussian white noise with a random intensity given by a critical real GMC. The
regions
${\mathcal{\mathcal{P}}}_{\mathrm{I/II}}^{}$
and
${\mathcal{\mathcal{P}}}_{\mathrm{II/III}}^{\prime}$
correspond to the phase boundary between the three different regions of the complex
GMC phase diagram. These results complete previous results obtained for the GMC
in phases I and III and only leave as an open problem the question of convergence in
phase II.
