Isolate workspaces in Gnome Shell, making the Overview look and behave as if the active workspace is the only workspace (except the workspace switcher).
This means it will only show an app icon in the dash if the application has a window on the active workspace (unless they are favourited), activating an application will try open a new window if there are no existing windows on the active workspace, and the overview will only display app icons as 'running' if the application has a window on the active workspace.

Original idea from: https://extensions.gnome.org/extension/876/isolated-workspaces/ and https://extensions.gnome.org/extension/440/workspace-separation-on-dash/

The previous approaches reimplemented many parts, causing incompatibilities and large codebases. This implementation is significantly smaller and easier to maintain as it does not reimplement half the Dash to achieve the same features. It also fixes a number of bugs, as well as adding new features. For example, application icons are also updated.

This extension is not actively maintained, so consider using https://extensions.gnome.org/extension/1583/worksets/ with the approach from my extension:
#24 (comment)

A new fork of this extension for Gnome 45+:
https://github.com/KSXGitHub/workspace-isolated-dash