Mail client that gives notifications for multiple accounts

I'm looking for an email client that can handle multiple accounts and can give you a notification when an email is received on any of the accounts, preferably without it being necessary for the client to be running.

The Win10 Mail app can do all this, but it seems to randomly decide when it's gonna send notifications and when it's not gonna send them.

I don't know about the mail client not being open. Normally you need the client running to receive a notification.

Try Trillian.