The complex Gaussian Multiplicative Chaos (or complex GMC) is informally defined as a random measure $e^{γX} \mathrm{d} x$ where $X$ is a log correlated Gaussian field on $\mathbb R^d$ and $γ=α+iβ$ is a complex parameter. The correlation function of $X$ is of the form $$ K(x,y)= \log \frac{1}{|x-y|}+ L(x,y),$$ where $L$ is a continuous function. In the present paper, we consider the cases $γ\in \mathcal P_{\mathrm{I/II}}$ and $γ\in \mathcal{P}'_{\mathrm{II/III}}$ where $$ \mathcal P_{\mathrm{I/II}}:= \{ α+i β\ : α,β\in \mathbb R \ ; |α|>|β| \ ; \ |α|+|β|=\sqrt{2d} \}, $$ and $$ \mathcal{P}'_{\mathrm{II/III}}:= \{ α+i β\ : α,β\in \mathbb R \ ; \ |α|= \sqrt{d/2} \ ; \ |β|>\sqrt{2d} \},$$ We prove that if $X$ is replaced by an approximation $X_ε$ obtained via mollification, then $e^{γX_ε} \mathrm{d} x$, when properly rescaled, converges when $ε\to 0$. The limit does not depend on the mollification kernel. When $γ\in \mathcal P_{\mathrm{I/II}}$, the convergence holds in probability and in $L^p$ for some value of $p\in [1,\sqrt{2d}/α)$. When $γ\in \mathcal{P}'_{\mathrm{II/III}}$ the convergence holds only in law. In this latter case, the limit can be described a complex Gaussian white noise with a random intensity given by a critical real GMC. The regions $\mathcal P_{\mathrm{I/II}}$ and $ \mathcal{P}'_{\mathrm{II/III}}$ correspond to phase boundary between the three different regions of the complex GMC phase diagram. These results complete previous results obtained for the GMC in phase I and III and only leave as an open problem the question of convergence in phase II.