Thanks @Limi for the suggestion!
We have an existing feature request here, which I encourage you to upvote and track its progress.
On a side note, we’re considering to add a “teams sidebar” for people who belong to multiple teams, so you can see notifications across multiple teams easier:
You may also consider using the Mattermost Desktop Apps, which supports connecting to different teams across multiple Mattermost servers from a single interface, and switch to them with shortcut keys.