I could not find anything on the internet, so I hope someone here has an awnser to my question.
Is it possible to extend the REST API for Magento and if so, does someone have an example or guide on how to do this?
The Magento REST Api can indeed be extended. There is a blog post on my website showing how to make a start on doing this http://web.archive.org/web/20130512072025/http://magepim.com/news/Extending-the-Magento-REST-API-part-1_13
It only covers the basic setup but hopefully it'll give you a starting point.