You will need the specific approval of NZCAA to fly the aircraft in NZ, as an Australian sub-ICAO Annex VIII C.of A.(any Limited, Restricted or Experimental) is not valid outside Australia territorial limits.
Do all you homework carefully, especially the NZ tax laws. Some of the valuation for tax are mind blowing.
NZCAA are easy people to work with, quite pleasantly different to CASA, and the fees are very moderate.