Since you are trying to maximize something that has a large complicated denominator...

I wondered what happened if tried to minimize the square of that denominator

Minimize[Denominator[Together[-(b/(3 a))+((1+I Sqrt[3])(-b^2+3 a c))/
  (3 2^(2/3)a(-2 b^3+9 a b c-27 a^2 d+Sqrt[4(-b^2+3 a c)^3+(-2 b^3+
  9 a b c-27 a^2 d)^2])^(1/3))-((1-I Sqrt[3])(-2 b^3+9 a b c-27 a^2 d+
  Sqrt[4(-b^2+3 a c)^3+(-2 b^3+9 a b c-27 a^2 d)^2])^(1/3))/(6 2^(1/3)a)]]^2,{a,b,c,d}]

which quickly returns

{0, {a -> -1, b -> -1, c -> -1/3, d -> 0}}

I believe you have to be careful using that information. I don't think you can just plug those values into your original expression. But with care I think you might be able to infer your maximum may be infinity.

Please check all this VERY carefully before you even think of trusting it.