You Do Need power plug adapter while traveling from Saint Barthélemy to New Zealand.

Here is Why You Need An Adapter

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

Saint Barthélemy

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

Type C

Type E

New Zealand

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

Type I