Q:

How to express a function in terms of new variable?

I have 
In[57]:= g[a_, x_, y_] = 
 Simplify[Limit[f4[t], t -> Infinity]] + 
  Sqrt[-1]*Simplify[Limit[f5[t], t -> Infinity]]

Out[57]= ConditionalExpression[(1 + a + a x + a^2 x - a y)/(
  3 + 3 a + a^2) + (I (1 + a x + 2 a y + a^2 y))/(
  3 + 3 a + a^2), (x | y) \[Element] Reals && a > 0]

Now I want to express the g[t,z] in terms of a new variable
z = x + Sqrt[-1] y;

I was looking for expression like this

A:

Perhaps that's the answe to your question 
expr=(1 + a + a x + a^2 x - a y)/(3 + 3 a +a^2) + (I (1 + a x + 2 a y + a^2 y))/(3 + 3 a + a^2);
expr /. {x -> Re[z], y -> Im[z]} // FullSimplify
(* ((1 + I) + a + a ((2 + I) + a) z - a Re[z])/(3 + a (3 + a))*) 

