You can have all your iCloud Mail email forwarded automatically to another email account. You may find this helpful if you want all your iCloud email in your new branded email account.
Forward all messages automatically
In iCloud Mail, click the Action pop-up menu gearbox in the sidebar, then choose Preferences.
In the General pane, select “Forward my email to,” then type your new email address in the text field.
If you don’t want to store a copy of each forwarded message in iCloud, select “Delete messages after forwarding.”