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For q > 0 q>0 let A \mathcal {A} denote the unital ∗ * -algebra with generator x x and defining relation x x ∗ = q x ∗ x xx^*=qx^*x . Based on this algebra we study q q -normal operators and the complex q q -moment problem. Among other things, we prove a spectral theorem for q q -normal operators, a variant of Haviland’s theorem and a strict Positivstellensatz for A . \mathcal {A}. We also construct an example of a positive element of A \mathcal {A} which is not a sum of squares. It is used to prove the existence of a formally q q -normal operator which is not extendable to a q q -normal one in a larger Hilbert space and of a positive functional on A \mathcal {A} which is not strongly positive.