Galois group of quartic with only complex roots

Let us view the Galois group $G$ as a group of permutations on the four roots.

No, the fact that the roots form two complex conjugate pairs does NOT let us conclude that the group has no elements of order three (I will produce an example if necessary). All it implies is that the group $G$ contains a product of two disjoint 2-cycles.

What is confusing you is probably the fact that the complex conjugation does not necessarily belong to the center of $G$. In other words, there may be automorphisms $\tau$ that don't commute with complex conjugation. For such an automorphism $\overline{\tau(z)}$ may be different from $\tau(\overline{z}).$