Is there a way to display desktop notifications even when Chrome or Firefox is closed?

If you have a "background" permission in manifest.json, your background page will be able to show notifications even when Chrome window is closed.

"permissions": [
    "background"
],

As stated in the documentation:

When any installed hosted app, packaged app, or extension has "background" permission, Chrome runs (invisibly) as soon as the user logs into their computer—before the user launches Chrome. The "background" permission also makes Chrome continue running (even after its last window is closed) until the user explicitly quits Chrome.

You need to use the "background" permission with a background page, event page or a background window for hosted apps.

For web, use Push API for Chrome and other browsers. The advantage of using push messages is that even if your page is closed, your service worker will be woken up and be able to show a notification. Web Sockets and EventSource have their connection closed when the page or browser is closed so it's not recommended. Here is the documentation and example.

In Chrome. Only those users that happen to have an extension installed that requires background mode, like hangouts, will be able to receive push notifications when chrome is not "running". It does not seem like a good idea to rely on it.

The chrome team seems to be considering it for web push too but so far there is no ETA.

https://code.google.com/p/chromium/issues/detail?id=402456