I've played about 50 games and haven't built/seen a single Archon. When should they be used?
One possible use case is when you're massing Dark Templars, and then find that your opponent has detection everywhere.
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In general though, your high templar are going to be more effective vs most things, so you'll usually only morph them after storming, but try them out in some of the scenarios I listed, they are actually fairly powerful.
There's also the Archon toilet, wherein you throw 5-6 Archons in a Vortex along with your opponent's army. Since Archons do splash damage, they'll do incredible damage when they pop out of the Vortex.
Though you can probably only pull this off in a team game, since it requires both a Mothership and a lot of Archons.