How to use another email address to receive message notifications?

If you want to use a specific email for alerts and updates, you can create an Attorney account using that email.
For better understanding, let's say your designated email is [e-mail].
Step 1. Turn off email notifications of your current account. (if you do not wish to receive any system generated email notifications)
Go to Account Setting.
You will see 5 types of notifications.Change all preferences to No email notification.

Step 2. Create an Attorney account using [e-mail]
Go to Team Account=>Account profiles=>+ Create
Step 3. Put in information and set the role as Attorney. Click Create.
Step 4. Assign cases to both accounts
After adding a case file, get into case details and go to Checklist. Assign the case to both accounts and click Save.
Step 5. Log in to your newly created account ([e-mail]) and enable all email notifications according to step 1.
Once you have finished the above setting, both accounts will receive case related notifications in the Notification Center. (Bell Icon)
In addition to the onsite alerts, your account will receive no email notifications (you turned it off in step 1) and [e-mail] will receive all email notifications.