The creation operator has no eigenstates. But one postulate of QM says that the state of a system after measuement using an operator must be an eigenstate of that operator. How to make sense of this?
Not all operators are observables. Only self-adjoint operators are observables. A property of self-adjoint operators is that they have real eigenvalues.
The creation operator is not an observable and is not a self-adjoint operator. It's okay for an operator not to have eigenstates and not to be an observable.
Only observables can be measured.