My personal preference would be to have a notification sent to the default support address, but sending an email to the affected Gmail account is an acceptable second choice (we will just create a filter to forward this notification to our default address).
The reason I suggest sending the notification to the default support address is because chances are if users have connected a Gmail account, they are not checking it and instead rely on Zendesk to do so.
The 'pie in the sky' feature request would be to have a pop-up notification in Zendesk itself. Given the importance of connections to external support addresses, this seems pretty reasonable!
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