Suspension & Brakes - Break Booster or Master Cylinder?

When the car is off the pedal is loose not stiff.. when I start the car and pump the breaks, the pedal will get stiff.. When I drive the car and as i come to a stop the break pedal will go down halfway an then drop down about another inch until it reaches the back wall..

Anyone have an idea?

Sounds more like the master cyl. When the booster goes bad, it usually creates a hard pedal symptom.

I would check all your brake lines, connections and calipers for any new signs of fresh brake fluid before replacing your master cyl. though.