\begin{equation} f(x) = e^{ix} = \cos (x) + i\sin (x) \end{equation}

this brings a circle of radius one, because in every point, velocity is orthogonal to where you are (because \(f’(x) = if(x)\), and multiplying by \(i\) accounts for a rotation of 90 degrees.

And so,

\begin{equation} z = re^{i\theta} \end{equation}

gives any point in the imaginary polar plane.