as a prelude to inquiring about solutions of Pythagoras' equation in Gaussian integers, it seemed sensible first to write out this equation for the complex case! i use the notation $z_i=x_i+iy_i$ and denote by $z^*$ the complex conjugate of $z$ $$ z_1^2 + z_2^2 = {z^*}_3^2 $$ or, in components: $$ x_1^2+x_2^2+y_3^2 = y_1^2+y_2^2+x_3^2 $$ and $$ x_1y_1 + x_2y_2 +y_3x_3 = 0 $$ in other words if $u,v \in \mathbb{R}^3$ with coordinates: $$ u =(x_1,x_2,y_3)\\ v =(y_1,y_2,x_3) $$ satisfy $$ \| u \| = \| v \| $$ and $$ u \cdot v = 0 $$ then we may derive from the coordinates (in the manner indicated) three complex numbers satisfying the Pythagorean equation.
question what is the correct geometric/algebraic interpretation of this fact? it may be of no particular significance, but the two little twists necessary to achieve notational felicity suggest there may be something sinister going on in the background.
grateful for any insights...