I am pleased to inform you that desktop notifications exist for Inbox by Gmail as well. I have scoured the Chrome Webstore to try and find all of the extensions that provide notification, or similar, functionality. Below is the list with items I use marked with (*).
Thanks to +Tom Voute for bringing this to my attention: "It should be noted that all these notifiers are actually Unread counters of the Gmail API hence also notify for bundles whose notifications have been turned off in the mobile apps since you can't yet turn off notifications for specific bundles in the web version. For example, if you have a bundle set to show daily at 7:00 AM, the notifier will show an unread message immediately upon entering your inbox. Currently the only way to get a notification that honors this setup is to use Pushbullet."
- (*) Google Inbox Checker (Inbox by Gmail)
- Google Inbox Checker (Inbox by Gmail)
- (*)Gmelius for Inbox by Gmail This provides an unread count in the favicon in the browser tab
- Inbox by Gmail Checker
- Mail Checker for Google Inbox
- Checker Plus for Gmail - added several features to support Inbox by Gmail.
If you have found more, please let me know, and I will add to the list.