I wondered what happens when you try to do this in more than 2 dimensions? It turns out that you can just extend the negation of the imaginary part before proceeding.

https://math.libretexts.org/Bookshelves/Abstract_and_Geometr....

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.

zz^* is fine for scalar complex numbers, but z^*z is nice because it also works for vectors. You can think of the complex conjugate as a special case of an adjoint, and the hermetian transpose is another special case.

zz* !? Why not z*z if you're nitpicking already? inb4 WFH killed Commutators.