You Do Need power plug adapter while traveling from Norfolk Island to Saint Barthélemy.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Barthélemy are C and E and in Norfolk Island is I. This is how they look like.

Norfolk Island

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

Type I

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E