Hectic absolute values? (where $a=ix$ and $b=-ix$) Where $a=ix$ and $b=-ix$ then what is: $$|a+b|^2$$ $$|b-a|^2$$ And then is this equality true? $$|a+b|^2=|a|^2+|b|^2$$ because it seems $a+b=0$!

You have $a+b=0$ and $b-a=-2ix$ so $|a+b|=0$ and $|b-a|=2|x|$ Since $|a+b|^2=0$ and $|a|^2 + |b|^2 = 2|x|^2$ you have not this équality in general. The general formula about complexe numbers is : $$|a+b|^2=|a|^2 + |b|^2 + 2 {\mathcal R}e(\overline ab)$$