For Fastmail (current web interface) this works fine with Firefox 48.0.2 (and other versions since 44) on Windows. The Fastmail web interface for you account needs to be running in some tab, but you do not need to be viewing that tab. The pop-up appears in the lower right corner of my primary display for about 12 seconds (set by Firefox). Here is what this looks like on my PC (in the extreme lower right above the taskbar):
http://www.mediafire.com/view/jugg5g...ification.png#
For a description of this feature, see:
https://support.mozilla.org/en-US/kb...urce=inproduct
To be sure that Firefox notifications for Fastmail are enabled, enter
about:preferences#content in the URL bar (or go to
Options>Content). In the Notifications area, be sure that
Do not disturb me is not enabled, then click
Choose... on the right and be sure that
https://www.fastmail.com is allowed.
The quick (12 seconds with Firefox 48.0.2) disappearance of these notifications bothers many users, and there is a project underway to fix this in some way. See;
https://bugzilla.mozilla.org/show_bug.cgi?id=862395
Bill