You Do Need power plug adapter while traveling from Cayman Islands to Norfolk Island.

Here is Why You Need An Adapter

The outlet and plug type used in Norfolk Island 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

Norfolk Island

In Norfolk Island, outlets of type I are in use.

Type I