Unfortunately you have run into a shortcoming of Mathematica. FindFit
seems to have been programmed with the real case in mind. The complex case works in some simple examples such as
FindFit[{I, 2, 3}, a*x , {a}, x]
FindFit[{1, 2 + I, 3}, a *x, a, x]
FindFit[{1, 2, 3}, a x + I , a, x]
but it fails with little variations:
FindFit[{I, 2, 3}, Cos[a] , {a}, x]
FindFit[{1, 2 + I, 3}, {a *x, {0 < Re[a] < 1}}, a, x]
FindFit[{1, 2, 3}, {a x + I, {0 < a < 1}} , a, x]