Isn’t the squaring actually multiplication by the complex conjugate when working in the complex plane? i.e., √((1 - 0 i) (1 + 0 i) + (0 - i) (0 + i)) = √(1 + - i2) = √(1 + 1) = √2. I could be totally off base here and could be confusing with something else…
Good rule of thumb: if someone else hasn’t solved the problem yet, it’s more complicated than you’re assuming. If the problem is worth solving, other people smarter than you have almost certainly attempted the easy “solutions” already, and they were inadequate to solve the problem. Heck, even if it’s not worth solving, there’s a non-zero chance that some pre-Reagan weirdos took a crack at it with bonus mercury and thallium compounds for the lulz and published it all in a vague 200-word comm in a now-defunct journal.