How about the following idea for representing complex intervals: _conic sections_.
The problem I'm trying to solve is the following one:
How do you provide a set of atomic intervals that is expressive enough to handle both real intervals -- e.g., [3,5] -- as well as circular regions in the complex plane -- e.g., {x | (x-C)<r^2}.
Possible answer: conic sections, including degenerate conic sections.
The interval [3,5] is a degenerate ellipse with foci at (3,0) and (5,0), where the string is completely taut.
The interval [3,INF) is a degenerate parabola with the focus at (3,0) and stretching to the right; (MINF,3] is the degenerate parabola with the focus at (3,0) stretching to the left.
The interval [3,3] is a degenerate ellipse with foci at (3,0) and (3,0), or alternatively, a circle with center (3,0) and radius=0.
One tradition extension of intervals to the complex plane uses _circles_ (Henrici's "circular arithmetic"), so we can represent these.
So far, we have used only conics with eccentricities <=1. I don't know if there are any useful "intervals" which are hyperbolic.
I don't know enough about elliptic functions to know if elliptical regions in the complex plane are particularly useful.