In the old Outlook.com you would configure your business email address like myname @ mydomain.com to forward emails to your Outlook.com account and as a “send only” account so you could send emails from that address.

After the forced migration to the new Outlook.com this feature has been removed. The only way to send emails on your business email address is to set up a POP account (IMAP does not work reliably at this time).

However Outlook.com only collects mail every 30 mins so that is very bad. To solve this problem,

You login to your email account at myname @ mydomain.com and,

1. create a folder called _inbox_sweep
2. create a rule to
   1. Forward incoming mail to myname @ outlook.com and,
   2. to move the mail to the _inbox_sweep folder

This will then ensure that your receive emails immediately and you do not get duplicate emails when Outlook.com checks the POP account.

If you do not want a backup and you don’t want your business mailbox to fill up you can set 2.2 above to delete but do a few days testing first.

I have cross-posted this on the Microsoft Community site and there may be some comments there.