$\frac{a+bi}{c+di} = \frac{a+bi}{c+di} * \frac{c-di}{c-di} = i (\frac{b c}{c^2+d^2}-\frac{a d}{c^2+d^2})+\frac{a c}{c^2+d^2}+\frac{b d}{c^2+d^2}$. At this point, you should be able to get the magnitude easily. Yes, it'll be cumbersome computation wise, but that should be it.
Suppose $e = \frac{b c}{c^2+d^2}-\frac{a d}{c^2+d^2}$ and $f = \frac{a c}{c^2+d^2}+\frac{b d}{c^2+d^2}$
Then, $\|f + ei\| = \sqrt{f^2+e^2} = \sqrt{\frac{(bc-ad)^2}{(c^2+d^2)^2} + \frac{(ac+bd)^2}{(c^2+d^2)^2}} = \sqrt{\frac{2(a^2d^2+b^2c^2)}{(c^2+d^2)^2}}$ and you could take it from there.