While not available across the board -- meaning, you won't be able to do this once, but have to do it for every Account -- you can create Events for users that can send an email to a separate address on "message received". That way, for any message they get to their business account, that notification email can fire. 

 

System variables are available as well, so that notification email can contain the sender's info, the email subject, etc. That may help the end user determine if it's something they need to look at right away or if it can wait. You can also customize the email's subject, so using "New email received at your Company Name account" would work. (Unfortunately using system variables in the subject isn't available at this time.)

 

Also, this is really a question since it's possible to do what you want, so I'm going to shift the category...