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

Here is Why You Need An Adapter

The outlet and plug type used in Bhutan are C, D and G and in New Zealand is I. This is how they look like.

New Zealand

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

Type I

Bhutan

In Bhutan, outlets of type C, D and G are in use.

Type C

Type D

Type G