Unique properties of pure Imaginary numbers?

Are there any non trivial properties unique of the imaginary numbers?

By trivial I mean stuff like $\bar a=-a$.

• It's still not clear to me what would qualify as non-trivial. – Michael Albanese Apr 19 '15 at 22:07
• For example $z^2 \in (-\infty, 0)$? – Robert Israel Apr 19 '15 at 22:10
• Like $\lvert e^a \rvert = 1$ precisely when the real part of $a$ is $0$? – pjs36 Apr 19 '15 at 22:11
• The title says "pure imaginary numbers" but the body of the question says "imaginary numbers". Which is it? – bof Apr 19 '15 at 22:53
• @bof Pure I meant a complex number that has real part = 0. – YoTengoUnLCD Apr 19 '15 at 22:54

$$|e^z| = 1$$ for $$z$$ purely imaginary.