Solving complex equation $z^2 + (1+i) \overline{z} + 4i = 0$ Consider the following equation, where $z \in \mathbb{C}$, $i$ is the imaginary unit and $\overline{z}$ is the conjugate of $z$:
$$
z^2 + (1+i) \overline{z} + 4i = 0
$$
What is the method to deal with equations such as this?
I have tried various things: I tried substituting $z$ with $a+bi$, or $re^{i\theta}$, hoping I'd notice something. I thought I could somehow transform this into a quadratic equation, but I couldn't. Now I have no idea what to try. I'd appreciate ideas greatly.

$$a^2-b^2+2iab+(1+i)(a-ib)+4i=0\\ a^2-b^2+a+b+(2ab+a-b+4)i=0\\ \begin{cases}a^2-b^2+a+b=0\\ 2ab+a-b+4=0\end{cases}\\ \begin{cases}(a+b)(a-b+1)=0\\ 2ab+a-b+4=0\end{cases}\\ \begin{cases}a=-b\\ -2a^2+2a+4=0\end{cases}\vee\begin{cases}b=a+1\\ 2a^2+2a+3=0\end{cases}$$
An these can be solved for $a,b\in\Bbb R$ accordingly.
