Draw $\{z\in\mathbb{C}|z\overline{z}<3+2\text{Im}(z)\}$

Question

I want to draw the set $\{z\in\mathbb{C}|z\overline{z}<3+2\text{Im}(z)\}$. However, I don't know what to do about $2\text{Im}(z)$. If the set would be $\{z\in\mathbb{C}|z\overline{z}<3\}$, it'd be quite easy, since $z\overline z=\vert z\vert^2$, so the set would contain all complex numbers inside the circle around the center with radius $r=\sqrt{3}$. But how to interpret $2\text{Im}(z)$?

Answer 1

Write this as $x^2+y^2<3+2y\iff x^2+(y-1)^2<4$ which is just the interior of the radius $2$ disc centered at $(0,1)$.

Answer 2

Since $\operatorname{Im}z = \frac{1}{2i}(z - \bar z)$ the condition can be rewritten as:

$$ z \bar z \lt 3 - i(z-\bar z) \\ z \bar z + i z - i \bar z + 1 \lt 4 \\ (z - i)(\bar z +i) \lt 4 \\ |z-i|^2 \lt 4 $$

Therefore the set is the interior of the disc of radius $2$ centered at $i$.