Complex Algebra

Note first that $z$ cannot be purely imaginary, since then $z^{2} + 1$ would be real, making $w$ imaginary. So we know that $z = a + bi$ for non-zero reals $a,b$ .

Assuming that $w,k$ are non-zero as well, we have that

$w(z^{2} + 1) = kz \Longrightarrow z^{2} - \dfrac{k}{w}z + 1 = 0 \Longrightarrow z = \dfrac{\frac{k}{w} \pm \sqrt{\left(\frac{k}{w}\right)^{2} - 4}}{2} = \dfrac{k}{2w} \pm \sqrt{\left(\dfrac{k}{2w}\right)^{2} - 1}$ .

Now as $Im(z) \ne 0$ we must have $\left(\dfrac{k}{2w}\right)^{2} - 1 \lt 0$ , so

$z = \dfrac{k}{2w} \pm i\sqrt{1 - \left(\dfrac{k}{2w}\right)^{2}} = a \pm i\sqrt{1 - a^{2}}$ where $a = \dfrac{k}{2w} \ne 0$ .

Then $|z| = \sqrt{a^{2} + (1 - a^{2})} = \boxed{1}$ .