That’s fairly simple: we restrict the complex phase to the range (-pi, pi] and the principal square root halves the complex phase. -1 has the phase value pi, so the principal square root has the the complex phase pi/2, so it’s i, while -i has a phase of -pi/2

This is technically incorrect. While the square root is both the positive and the negative solution, the sqrt (√) operator results in the principal square root. For nonnegative numbers this is the nonnegative square root and more generalised for complex numbers it’s the square root that halves the complex phase.

Now I am confused. How is engineering disjunct from science? And even beyond that, there is astronomy involved (star map, eurion being based on Orion), which has something to do with science.

The eurion constellation is a security feature used on many banknotes. It’s made up from 5 circles and often integrated into larger patterns that make it less obvious. Printers identify it and reject to print money.