  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.
