Can Fermat's last theorem be proven false with complex numbers?

Question

Fermat's famous last theorem (a^n + b^n ≠ c^n, for n > 2) so far holds true.....with real numbers. I was wondering if it could be false if one extended it to complex numbers, that is, (a+bi)^n + (c+di)^n = (e+fi)^n... I have tried to weaken the problem by assuming that n is 3 and that (a+bi) = (c+di) (for the Pythagorean theorem, it is valid to have two sides of the same length, and if one can prove the theorem false for two of the same complex number, it is still a valid counterexample). Knowing that, I expanded 2(a+bi)^3 = (e+fi)^3 out into a nightmarishly long cubic equation, but now I am stuck because I am not sure how to simplify or combine both sides or even how to go about it. I did take a look at the complex plane, and tried cubing the simple complex number 2+i, which turns out to be 2+11i. Then I picked another complex number, cubed it, added the two numbers together and I got a number that does not have an integer cubed root. That was just trial and error, though, not a very mathematically sound method, just looking for any patterns. My hope is that if the theorem can be proven false for n=3, can it be for 4? 5? Any other value of n?