You Do Need power plug adapter while traveling from Anguilla to Cocos (Keeling) Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cocos (Keeling) Islands is I and in Anguilla are A and B. This is how they look like.

Anguilla

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

Type A

Type B

Cocos (Keeling) Islands

In Cocos (Keeling) Islands, outlets of type I are in use.

Type I