When Births, Deaths and Marriages has officially registered the death they’ll issue a Death Certificate (officially this is called a “New Zealand Death Certificate”).
Having a copy of the Death Certificate can be important, for example, for getting funds released from the deceased’s bank when it’s only a small estate (see in this chapter “Dealing with the deceased’s property / Small estates: No need for court approval”).
To get a copy of the certificate you’ll need to apply for it, and pay the relevant fee. You can apply when you fill out the notification form (see above).