Managing desktop notifications from multiple accounts on one Outlook app

Follow these steps and see if it will work for you:

How do I turn off Desktop Alerts for an account that I own that isn't my primary email account in Outlook?

You have to disable Desktop alerts in Options, and then create a rule to display alerts on the primary account. 

First you need to disable the Desktop Alerts in Outlook.

1. Click File
2. Click Options
3. Click Mail
4. In the Message Arrival section, uncheck Display a Desktop Alert
5. Click OK

Then, create a rule in your email account to display alerts for the account you want displayed. 

1. Click File
2. Click Manage Rules and Alerts
3. Make sure your inbox is selected in Apply changes to this folder
4. Click New Rule...
5. Choose Apply rule on messages I receive and then click Next
6. Uncheck any conditions and click Next.
7. Click Yes to the warning about the rule being applied to every mail message received.
8. Click "display a Desktop Alert" and click Next.
9. Click Next.
10. Give it a name if you want and click Finish.