Only notify me when a new chat conversation begins, not for each message

Currently I am notified every time a chat message comes in, which means I get a ton of notifications all the time. I'd prefer to only be notified when a new chat conversation comes in (once I know there's a new chat conversation, I tend to stay on the conversation until it's resolved so I don't need subsequent notifications).

Ideally this notification could also be more visual (like a pop-up) since I am often on another application and I tend to tune out sound-only notifications.