This equation has roots $z_1$ and $z_2$ given by

${\displaystyle \frac{-p\pm \sqrt{p^2-4q}}{2}}$

so

${\displaystyle z_1=\frac{-p+\sqrt{p^2-4q}}{2}=u+\mathrm{iv}}$

and

${\displaystyle z_2=\frac{-p-\sqrt{p^2-4q}}{2}=u-\mathrm{iv}}$

where $u+\mathrm{iv}$ and $u-\mathrm{iv}$ are complex conjugates.

The product of the complex conjugate roots is given by

${\displaystyle z_1{z}_2=(u+\mathrm{iv})(u-\mathrm{iv})=u^2+v^2=|z_1{|}^2=|z_2{|}^2.}$

Hence as the quadratic changes keeping $q$ fixed and varying $p$ the locus of the complex roots is the circle with radius $\sqrt{q}$ centre at the origin.