I see that I can more or less easily create a custom second level domain aka contoso.cloudapp.net on Azure VM. But I would like to setup a real domain I purchased such a contoso.com without the clouapp.net.
Is that even possible with a Azure VM? Or do I need to purchase something else?
You have to use cname for you new domain to the cloudapp domain .
You can use an alias too . And there is a faq step by step to do that with azure gui or powershell tools .
http://azure.microsoft.com/en-us/documentation/articles/cloud-services-custom-domain-name/