We have a Remote Choice location guide for New Zealand. Check it out.
Entrepreneur Work Visa: You will have to invest NZ$100,000 (can be waived), score 120 points on the points scale, and present a business plan. If the business does well it can lead to a permanent visa.
Global Impact Visa: First you need to be accepted to the Edmund Hillary Fellowship.
"The Edmund Hillary Fellowship (EHF) fosters humankind's creative potential from New Zealand, by supporting a community of impact entrepreneurs, investors and changemakers to build meaningful solutions to global problems."
Once you're accepted you may apply for the permanent visa, after 30 months of being part of the fellowship.