= Solution
Put $z=x+iy$. The expression whose <complex argument> is constrained becomes
$$
z^2+1=(x^2-y^2+1)+2ixy.
$$
Its <principal argument> is $\pi/2$ exactly when its <real part> is zero and its <imaginary part> is \b[strictly positive]. Hence
$$
\boxed{y^2-x^2=1,\qquad xy>0.}
$$
The <locus> consists of the upper-right half of the upper <hyperbola> branch, $y=\sqrt{x^2+1}$ with $x>0$, and the lower-left half of the lower branch, $y=-\sqrt{x^2+1}$ with $x<0$. The vertices $z=\pm i$ are excluded because $z^2+1=0$ there and zero has no <complex argument>. The remaining halves have negative <imaginary part> and give argument $-\pi/2$, not $\pi/2$. The right panel of the original illustration marks the excluded vertices with open circles.
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