I have never heard of limit cycles bifurcated from real eigenvalues crossing zero. For the below system I believe that is the case. Is this bifurcation known and has a specific name?
For $d = b + i c \in \mathbb{C}$, $z(t) = x(t) + i y(t) \in \mathbb{C}$ and $a\in \mathbb{R}$, consider \begin{equation}\label{rb1} \dot{z} = a z + d z \vert z \vert^2 \end{equation} which can be written as \begin{equation} \begin{aligned} & \dot{x} = a x + (b x - c y) (x^2 + y^2)\\ & \dot{y} = a y + (b y + c x) (x^2 + y^2) \end{aligned} \end{equation} or in polar coordinates $z = r e^{i \theta}$, \begin{aligned} & \dot{r} = a r + b r^3 \\ & \dot{\theta} = c r^2 \end{aligned} For $a < 0$ the solution $z=0$ is stable. For $a > 0$ it is unstable. So there is a bifurcation at $a = 0$. If $c \ne 0$ and $a b < 0$ then there is a limit cycle $r^2 = - a/b$ which is stable if $b<0$ and unstable otherwise. The limit cycle is given by $$ z(t) = \sqrt{\frac{-a}{b}} e^{- i \frac{a c}{b} t } $$ Also the period of the limit cycle goes to infinity as $a \downarrow 0$. So unlike the Hopf bifurcation case, this limit cycle is very slowly rotating.