You Do Need power plug adapter while traveling from Cayman Islands to New Zealand.

Here is Why You Need An Adapter

The outlet and plug type used in New Zealand is I and in Cayman Islands are A and B. This is how they look like.

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B

New Zealand

In New Zealand, outlets of type I are in use.

Type I