Australia PR holder can work and live in New Zealand. The NZ resident visa also allows you to visit and study using your Australia PR.
The work in NZ is allowed for indefinite amount of time meaning that there are no time restrictions even with Australia Resident Return Visa.
You can only get the NZ resident visa on arrival. You cannot apply for it before your arrival.
New Zealand immigration uses your passport to check your Australia PR status. As a matter of fact, your Australia permanent residency is tagged to your passport when you first enter and validate the PR.
Starting 1 Oct 2019, if you are an Australian permanent resident you must hold an NZeTA (New Zealand Electronic Travel Authority) before you travel.
Australia Temporary Resident Visa Holders
Note that Australian temporary residents are not eligible for working in NZ. If you are in Australia on a specific work visa other than PR, you do need to apply for a Kiwi work visa to be eligible to work.
You can only apply for NZ resident visa for work when you arrive in New Zealand. You can’t apply for it before you travel.
Your resident visa will expire if you leave New Zealand.
New Zealand Income Taxes
You will need to pay New Zealand income taxes if you are working in NZ using your Australian PR.
Australia PR 189 – New Zealand
If you have the Australia 189 visa, then you are allowed to search for jobs in NZ and move to work if you get the job.
Remember that for your Australia PR to be valid, you should actually spend at-least 2 years in Australia to be eligible to get Australian resident return visa. If you are living and working in NZ, you will not be able to meet the Aussie’s PR requirements and may not be able to renew it.
Australia PR 190 Visa work in NZ
In case of Australia 190 Visa Residency, you have to stay and work in specific Aussie state region for at-least 2 years. If you do not, your PR may be cancelled.
Buying House in NZ on Aus PR?
You can buy house in New Zealand while you are working in New Zealand.
Source: NZ Immigration