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

Here is Why You Need An Adapter

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

Zambia

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

Type C

Type D

Type G

New Zealand

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

Type I