I can't prove that this group is not bounded (each cyclic subgroup is obviously bounded). But it seems that for general $A$ (when $A$, $A^{-1}$ don't have common eigenvalues) there are not many bases in which A is unitary.

What is the 3-dimensional example?

Fedor Petrov, thank you for noting this! The hard part is to prove that the operators are bounded.