I am wondering if there exists a different presentation of the complex number where the imaginary line from $-\infty$ to $+\infty$ for a given real part $x$ of $z=x+iy$ is represented by the circumference of the circle whose radius is $x$.
In the figure, the new plane (to the right) is a transformation of the ordinary complex plane with all $y$s lines are represented by the circumference of the circles whose radius are $x$s. In other words, the $y$-coordinate is now the angle-$mod-N$ where N=$2\pi x$ and $x$ is the radius.
For example, the point $z=5+i4$ is now represented by a point on the circumference of the circle with $r=5$ and located at $4$ Mod (N) where N is $2\pi (5)$.