Express as function of $z: \sin(a^2+b^2)+\cos(a^2+b^2)i$

Question

Where $z = a + bi$

and

$$f(a+bi) = \sin(a^2+b^2)+\cos(a^2+b^2)i$$

How could I write this in terms of $z$?

$$f(z) = ???$$

FYI, this describes a spiral of some sort. I stumbled across it making some mathematical artwork. Extra thanks to anyone who knows what kind of spiral.

Answer 1

Given that $|z|^2= a^2+b^2$ $$f(z) = \sin(|z|^2)+\cos(|z|^2) i = i e^{-i|z|^2}$$

Answer 2

$$f(z) = \sin(|z|^2)+\cos(|z|^2) i$$