Hi Marc,
As per your description, seems you are referring to the browser version of Outlook for Office 365, not the desktop version of Outlook, correct?
If this is the case, as far as I know, you can display the mail notification by clicking the gear icon, then click options>settings>message options, tick the option of "Display a notification when a new email message arrives".
Please note this is the forum to discuss questions and feedback for Microsoft Office client. Better to post your question in the dedicated forum of
Office 365 community, where you can get more experienced responses:
http://community.office365.com/en-us/f/158.aspx
The reason why we recommend posting appropriately is you will get the most qualified pool of respondents, and other partners who read the forums regularly can either share their knowledge or learn from your interaction with us. Thank you for your understanding.
Regards,
Ethan Hua
TechNet Community Support