I was thinking the same as @Afraser.
If you are already living here as a kiwi, and have been here for some years, you would definitely receive the same treatment as Aussies, if you've joined Medicare.
However, even if you moved over here now, I believe there is a also a 'waiting' period of some years before being able to access any health care (unless it is an accident) ...
Check out what the NZ Health site says:
https://www.mfat.govt.nz/en/countries-and-regions/australia-and-pacific/australia/new-zealand-high-commission-to-australia/living-in-australia/daily-life-in-australia/healthAnd the Aussie site:
https://www.servicesaustralia.gov.au/reciprocal-health-care-agreements-visiting-from-new-zealand?context=22481Essentially, it says: The Reciprocal Health Care Agreement doesn’t apply if you travel to Australia for medical treatment.