Auckland Visa Registration Guide
There is no registration required once you are granted a visa — your passport is all the government needs to track your comings and goings, and all relevant information like your NZ address and contact information will be included on your application. When you begin the application process, you will also be given a case number so immigration workers can easily access your entire history.
When you move to New Zealand, immigration officials will ask for your contact details by mail so that they can continue to keep track of you and offer you resettlement services. However, there is no legal obligation to keep immigration apprised of where you are.