Just wondering... if your neighbour threatens to kill you - is that a justifiable reason to call the cops?
i'm not familiar with the legal system in new zealand
but in most states in the US, the threat of physical force by one person toward another is a crime
it's called "Assault" in many states - it's usually a misdemeanor crime, punishable by fine
if the offender displays a weapon of some sort then it's called "Aggravated Assault" - that's usually considered a felony, punishable by incarceration
since it is no longer a crime in progress, the first step would be to call (the non-emergency number) or visit your local police station to file a report documenting the incident that occurred
you will be given the choice of leaving it at that or signing a criminal complaint against the offender
if you choose the first option, you will be given an incident number and a copy of the report - this may be useful later if the offender continues to cause trouble
if you choose the second option, the incident will be turned over to investigators who will then speak to the parties involved and any other corroborating witnesses to determine if there is sufficient evidence to arrest the offender
after that it gets really complicated