Finding complex roots from set of non-linear equations in python

When I encounter this type of problem I try to rewrite my function as an array of real and imaginary parts. For example, if f is your function which takes complex input array x (say x has size 2, for simplicity)

from numpy import *
def f(x):
    # Takes a complex-valued vector of size 2 and outputs a complex-valued vector of size 2
    return [x[0]-3*x[1]+1j+2, x[0]+x[1]]  # <-- for example

def real_f(x1):
    # converts a real-valued vector of size 4 to a complex-valued vector of size 2
    # outputs a real-valued vector of size 4
    x = [x1[0]+1j*x1[1],x1[2]+1j*x1[3]]
    actual_f = f(x)
    return [real(actual_f[0]),imag(actual_f[0]),real(actual_f[1]),imag(actual_f[1])]

The new function, real_f can be used in fsolve: the real and imaginary parts of the function are simultaneously solved for, treating the real and imaginary parts of the input argument as independent.