My 2004 Mazda6 has been making me nervous for the past week or so. It makes a clunking noise after the breaks have been applied and the car is at a complete stop. It feels like something is still shifting forward when the body of the car is stopped. What could this be that is shifting? It just got a new fuel pump about a month ago, could it be something that the mechanics messed with while replacing that? ...or something more foreboding and expensive...?

 

Hi, I need more info, but a common cause on these cars are the brake pads moving in the calipers.
The noise was more noticeable if non Mazda brake pads are used.
There used to be modified pad shims available from Mazda dealers to stop this noise, I may even still have a few in my shed. Make sure the brake pads are a good fit in the calipers and that grease is applied to the top and bottom of each brake pad.
The noise l'm describing is a clunk from the brakes milli seconds before the car comes to a stop. if you move forward, press brake pedal, stop, then move backwards, press brake pedal, stop, the clunk happens in both directions.
Not sure if this is what you're experiencing, but its a common noise more so even on the RX-8's.
By the way your fuel pump is removed through a hole under the back seat, so this should not be an issue, unless the mechanic left a tool on your fuel tank and this is your noise! Lol
Hope this helps!