I don't see the necessity to involve complex plane here. And I think the post above mine was trying to find the total arc length, instead of arc length parameterisation of the path (Archimedean spiral).
You could use polar coordinate or Cartesian coordinate; either way you should find it quite easy to parameterise the path using angle, and you could find the following arc length - angle relationship:
where P is the pitch.
Now to find out arc length parameterisation we need to find $\theta(L)$, i.e. to invert the above relationship. However, I'm not sure if it's possible and this answer is the closest thing I've found:
How to place objects equidistantly on an Archimedean spiral?