Unfortunately, I don't think you'll get notifications unless Chrome is running. When you exit out of Chrome, practically all of the processes used to support it are stopped. That's a good thing; otherwise everyone would have hidden processes running in the background. It's the same for Outlook and other mail programs.
I have Gmail on my phone to give me those notifications, because on phones, apps never truly shut off all the way. Most apps still have processes that check for updates, including incoming messages. Windows 10 does have the mail app, which through some trickery can probably implement Gmail notifications, but it's a background app almost as heavy as Chrome; you may as well just keep Chrome open all the time.
1And why does it have to be Chrome? Can't you ask Dagobert for another piece of tiny software which does the job all the time? – Run CMD – 2016-05-13T18:11:21.683
You are not going to get notifications about Google Mail if you don't have a browser open or a third-party client designed to display the notifications. – Ramhound – 2016-05-13T18:33:50.433
@JUICED_IT Thanks for the link to the extension. It seems like it works when Chrome is on but Gmail isn't opened but not when Chrome isn't opened. Anyway, it's better than nothing. – Donald Duck – 2016-05-13T18:54:58.687
1@JUICED_IT I just noticed that if you go into the settings for Checker plus for Gmail, you can check the checkbox "execute in background when Google Chrome is closed" you can get notifications even when Chrome is closed. That answered my question. You should have posted the link to the extension as an answer so that I could accept it. – Donald Duck – 2016-05-13T19:04:11.537