Do I need an extra plan with my carrier for an Apple Watch 4?

A cellular Apple Watch MUST be connected as a second device on your iPhone plan. That’s how the watch gets to share your iPhone’s phone number. If you are on a supported cellular network (https://www.apple.com/watch/cellular) your network will know how to do this, usually in exchange for a small monthly extra fee.

Apple Watch carriers

https://www.apple.com/watch/cellular/

NOTE: if you always have your iPhone nearby, the GPS only model can do "Everything" the Cellular model can do. The Cellular model has the extra capability of making/receiving phone calls, text/messaging, streaming Apple Music, getting weather data, etc, while you are out for a run, bike ride, swim without your iPhone. But whenever either model Apple Watch is near its paired iPhone, they will use the iPhone radios to save the Apple Watch battery life.

Not exactly an extra plan but an add-on at typically $10/month.

I almost forgot... Not all carriers support the watch.