Goedel's Completeness Theorem is for semantic completeness, and still applies to first-order logic. It says, "every theorem which is semantically true for its theory (true in all models of that theory) is syntactically provable." Goedel's Incompleteness Theorem applies to syntactic completeness. Syntactic completeness would mean, "Every syntactically-valid formula is either provable or refutable." Instead, what Goedel's Incompleteness Theorem tells us is: "Some syntactically-valid formulas are neither provable nor refutable, because they're true in some models (standard models) but not in others (nonstandard models)."

Again, this applies to any given formal system. Goedel's Incompleteness Theorem is parametric over formal systems, with first-order Peano Arithmetic being one of the weakest, most standardized systems in which it applies. The real condition for the Theorem is, "Any formal system sufficient to describe Turing machines."

>Say system with new 'quantum hack' operator.

That operator is called a Turing Oracle, and it's physically impossible. Possessing a Turing Oracle is equivalent to reversing the Second Law of Thermodynamics and refuting Heisenberg's Uncertainty Principle. It's physically wrong.