What you can do is get a domain and then take any webhost and host your mail account there.
.co.nz is really expensive.The cheapest I found was AUD 19 per year.
You can get .com for cheap.
Then you can get a cheap webhost like http://kuzhost.com/ (I am Using this one.)
or http://hostingcare.in/linux_hosting.html ($14 a year for 5Gb Space and Unlimited Bandwidth)
As Mihir said, you can either pay for just email hosting, or you can pay for web hosting and use the email hosting that comes with that. Either way you'll need to purchase a domain name, but with web hosting, you can also set up a website on your domain, instead of only using it for email.
It is also very simple to forward your email to another address, for example I forward mail from 5 other webmail accounts to my Gmail account and handle it all from there.
These guys are correct, the best way to do this is to buy a domain name and then get some cheap web hosting.
I recommend the following ;
-get your .com domain from godaddy, have a look on fatwallet for the latest discount code. This should get you a .com domain for about £5.
-get a £10 yearly hosting package from hukhosting and you can set up their free email, mail forwarding and redirecting for free, they use the outitgoes mail server which can be viewed online from any PC/smartphone.