For Functions That Do Not Have Unique Values, mathematical chooses a specific value; when plotting, say, the eigenvalue of a matrix that has such branch cuts, if the eigenvalues do not cross each other, one can do a simple Sort like this one to get the correct color. However, consider this MWE:

z[t_] := Cos[t] + I*Sin[t] Plot[{-I - Sqrt[1 - 9 z[t]^2], -I + Sqrt[1 - 9 z[t]^2]} // Im //    Evaluate, {t, 0, T}] 

the result has some discontinuities, hence different colors contrary to what one may intuitively think as a continuous curve:

For analytical functions, I can manually change/rewrite the expression so that the color does not change midway, but my problem is the eigenvalues comes from random processes where z[t] is a complex-valued random function, and Sort may not be a good solution due to possible level crossings of the eigenvalues. Is there some way to solve this problem for arbitrary complex z[t]?