I don't think you're going to be able to solve this purely with Gmail. You're going to need to use a third-party service.
One thing that comes to mind is If This Then That.
You could set up a recipe like this:
Trigger (from the Gmail channel): "New email in inbox from"
(Alternatively, you can use the "New email in inbox labeled" or "New email in inbox from search" triggers.)
Action (from the Gmail channel): "Send an email" to
firstname.lastname@example.org: "There is a new message from 'email@example.com'"
If you want the email to come from IFTTT rather than your own Gmail address, you can use the Email channel.
Other options are: You don't need to use email at all for the action. You could use SMS, or Android notifications, or any number of other options. (For instance, I have a recipe that sends me an Android alert whenever I get an email messages from my wife.)
I don't work for IFTTT. I'm just an enthusiastic user.