One of my own domains is "4loeser.net" which I use, e.g., for this blog. To avoid collisions with some other services linked to this domain, I created a subdomain "bm" (as in BlueMix). Next I added a CNAME entry to route all requests for "bm.4loeser.net" to "mybluemix.net".
That was the easy part. The next is more complex because it requires two simple steps, not just one. In my Bluemix account I have to click on my email address in the upper right corner. The pulldown menu has an entry "Manage Organizations". Then, in the admin panel, you select "domains" and add your custom domain.
|Add a custom domain to Bluemix|
|Application settings in Bluemix|
|Route the Bluemix app to your domain|
Update: There is a follow-up article describing how I added a Cloudant / couchDB database to my Python application on Bluemix.