When I move to New Zealand, will I have dual passports?

It depends on the requirements of your existing country. New Zealand allows dual passports but some countries require the holding of only one passport. To begin with your status as a New Zealand Permanent Resident does not affect your existing citizenship as Permanent Residency is a visa fixed into your existing passport.

If after 5 years as a New Zealand Permanent Resident you wish to apply for New Zealand citizenship you will need to assess the requirements of your existing citizenship at that time.