= Solution
The <determinant> is $z\cdot z^3-(-i)(-i)=z^4+1$. Thus $z^4=-1=e^{i(\pi+2\pi n)}$, and its four <polynomial roots> are
$$
\boxed{z_k=e^{i(\pi/4+k\pi/2)},\qquad k=0,1,2,3.}
$$
In Cartesian form these are $(1+i)/\sqrt2$, $(-1+i)/\sqrt2$, $(-1-i)/\sqrt2$ and $(1-i)/\sqrt2$. Their <moduli> are one and their <complex arguments> are equally spaced by $\pi/2$. The <Argand diagram> is therefore a square centered at the origin, with sides parallel to the coordinate axes.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2019-quartic-root-square.png]
{title=The four roots of z to the fourth equals minus one, plotted on the unit circle in the complex plane.}
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