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You will need to apply to transfer your visa to your new passport.
The relevant Immigration NZ web page says this:
Working Holiday Visas, e-visas and other label-less visas which have been issued electronically (without a label) are recorded against the passport number at the time of your application.
If you get a new passport issued, you must inform us, so that your records can be updated, and you can continue to use the label-less visa for your travel to and from New Zealand.
The process for applying for a transfer is detailed on the same page. It's free for labelless visas.