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

Here is Why You Need An Adapter

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

New Zealand

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

Type I

Cayman Islands

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

Type A

Type B