To make Chrome drop you a notification on your phone when you get a mention, PMs, new topics or replies etc, do this:
Turn on your app notifications for Chrome in your settings
Click your avatar (top right) and then the cog logo
Go to ‘Preferences’ at the top
Go to ‘Notifications’ on the left
Click ‘Enable Notifications’
You should get a notification if you’re watching topics.
If the notifications piss you off, then unsubscribe for topics you aren’t interested in, or turn off the notifications for other sites (you can do this on a per-site basis in Chrome settings.