Should you pass a customer complain to the higher lever even you have solved and closed it ?

That depends on the complain. When a complain is closed is either because issue was resolved entirly or that particular incident. If a bug in the application caused this complain and might occur more in future then it must be reported to the higher level staff in a proper written reported with explaining the issue and suggested proper solution. Otherwise, the complain will be noted as closed handled task in the staff routine report.

A closed complaint can only be refered to top management unless there is need for management recommendation, else, there is no need as the complaint has been taken care of