Another way to think about it is that for a given z, we can write it as (r, θ) where r is |z| and θ is the rotation of that norm in complex space. The complex conjugate z̄ is (r, −θ). A product of complex numbers zw is (r, θ)×(s, τ)=(rs, θ + τ) so zz̄ = (r², θ − θ) = r² = |z|².