Just in case, anyone trying the most basic sample and seeing a corrupt basemap, here's an explanation and the solution.
Steps to reproduce
from arcgis.gis import *
gis = GIS()
gis.map()
Explanation
The notebook is running on an https (secured) connection. However, as we didn't authenticate to ArcGIS Online, we're getting the tiles (and some .js-code) from an http (non-secured) connection. Chrome and Firefox block that unsecured JavaScript code.
Workaround 1
Instead of using the secured https connection, use http://notebooks.esri.com instead. As everything will be loaded without encryption, Chrome / Firefox are happy with that.
However: If you change your script to an authenticated AGOL-Session, you will send your username / password in plain-text through the net!
Workaround 2
Both Chrome and Firefox have the option to explicitly allow loading unsecure content. In Chrome, there's a small icon on the right, in Firefox, it's right beside the https-indicator.
Thanks for the feedback. This has now been corrected. You should now be able to use https:// without needing a workaround.