Once you have decided to use the geometrical representation of the complex numbers, the justification of the norm of a complex number is Pythagoras's theorem.

Once you have decided to use the geometrical representation of the complex numbers, the justification of the norm of a complex number is Pythagoras's theorem. If you want a more "complex numbers" justification, you can say that you want to make the vector "real" by moving it to the real line, so you multiply it by the inverse rotation of the complex number that you are looking at. If you instead multiply it by the conjugate, then you need to compute the square root of the output since it multiplies by the radius again. I hope this helps!