I had the experience several years ago of working with a team that had developed a ternary computing system. It ran out of funding and was abandoned but I felt it was ahead of its time. Currently, what is the state of this (ternary computing) research and development? Is there a place online that one could suggest to look for more information?
Modular counting gates are probably the closest thing in complexity theory to what you're asking about. Modular gates sum their inputs and compare against 0 mod $p$. Many authors consider these gates as taking in values in the range $[0,p-1]$ since you can hook multiple wires between pairs of gates. This paper provides a good summary of results in the area up until its publication date (2010) and a result on probabilistically emulating the AND function with just modular gates in constant depth.
As far as it relates to your original question, much more focus is given to composite moduli of two distinct prime factors (e.g. 6) than to prime moduli such as 3, since the computational power of a composite modulus with distinct factors is greater and much less well understood, as the linked paper describes.