This work is a first step towards a theory of "$q$-deformed complex numbers". Assuming the invariance of the $q$-deformation under the action of the modular group I prove the existence and uniqueness of the operator of translations by~$i$ compatible with this action. Obtained in such a way $q$-deformed Gaussian integers have interesting properties and are related to the Chebyshev polynomials.

• 05A30 - \(q\)-calculus and related topics
• 05B99 - Designs and configurations
• 11B65 - Binomial coefficients; factorials; \(q\)-identities
• 11F06 - Structure of modular groups and generalizations; arithmetic groups
• 30E99 - Miscellaneous topics of analysis in the complex plane