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You can have a completed task disappear right away if you set that in the Preferences. If you don't set that, completed tasks will disappear when you change views (e.g., switch to another perspective), or when you click "Clean up."

Completed tasks are kept with the project and context they belong to, just not visible depending on your view settings. There's also a dedicated Perspective "Completed" to view all completed tasks over projects and contexts in the same window.