You Do Need power plug adapter while traveling from Cocos (Keeling) Islands 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 Cocos (Keeling) Islands is I. This is how they look like.

Cocos (Keeling) Islands

In Cocos (Keeling) Islands, 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