To help diagnose further and make sure it's the ABS module, have scanned the car for codes with VCDS or some other program? If so post the codes.
Other things to look for. If the ABS module is indeed faulty, you'll have strange activity on the instrument panel.
Flashing brake light, tach or speedo will read zero while car is running / driving.
Unplugging the ABS module will determine if the unit is faulty. If unplugging it makes the instrument cluster go back to normal operation, then you know for sure the ABS module is bad.
You listed a 2004 as model year, that will be a Bosch 5.7 ABS module.
These modules were prone to failure, a painfully known issue for Passat owners. Buying a used module is very risky considering the failure rate of these units.
This company repairs Bosch 5.7 modules for $150. You send yours in, they repair it and send it back to you. Pretty quick turn-around. I've used them myself.
ABS Module Repair