Similarly, see Pavel Grinfeld's argument [2] advocating the vector basis L = {1, x, x²-1/3} w.r.t. Legendre polynomials.
[1] https://en.wikipedia.org/wiki/Pavel_Grinfeld
[2] MathTheBeautiful: Why {1,x,x²} Is a Terrible Basis [video] https://www.youtube.com/watch?v=pYoGYQOXqTk&index=14&list=PL....
Donald Knuth developed the "quarter-imaginary number system" (based on 2i) for a high school talent search. It can represent every complex number with just the digits 0,1,2 and 3, without the use of a sign character.
Cool, a fellow PROMYS alum! The only application I can think of is to use the patterns to generate fractals graphics e.g. http://demonstrations.wolfram.com/NumberSystemsUsingAComplex...
I recommend base 5/2 + sqrt(3)i/2. You can write points in the plane using 7 possible 'digits' at each place. http://fractalcurves.com/Root7.html
[1] https://www.promys.org/