You Do Need power plug adapter while traveling from Switzerland 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 Switzerland are C and J. This is how they look like.

Switzerland

In Switzerland, outlets of type C and J are in use.

Type C

Type J

Saint Barthélemy

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

Type C

Type E