In rational CFT one considers a deformation parameter q which is a root of unity in the complex plane. For q = exp(2.pi.i/N), the basic case, this depends only on the positive integer N. The same N labels a triple of points (0,q,oo) on the Riemann sphere, which can be used to cover moduli, described by the q=1 case. And before one knows it there are modular tensor categories, Galois groups and all sorts of other goodies floating around, which might explain why Terence Tao has been interested in physical distance scales recently.
Brannen has looked at different scales in the Standard Model with such a varying c. If c was supposed to be the speed of light one might equally ask about the domain c > 1, which has of course been considered by Riofrio. So c could be very, very big, or it could be very, very small.