Typically, if a polynomial Julia set has non-empty interior, then there will be an attractive periodic orbit. Thus, the interior can be colored in much the same way that the exterior is colored - i.e., the color can be based on the number of iterates it takes to get close to the attractive orbit.
This is exactly the procedure that Wolfram Alpha uses to generate "escape time" Julia sets. By default, the Julia set is displayed using an inverse iteration algorithm, but there is a toggle switch that allows you to change to an escape time image, to get an image like so:
While this is not a color image, colors easily could have chosen, rather than shades of gray. One advantage of this approach is that it is equally applicable to rational functions, as well as polynomials. For example: