The TC ( all TC's, the "Q" body ) use a Teves anti lock brake system that is unique to the TC but is similar to the Teves systems used on other European cars ( and possibly the Buick Grand Nationals)
The black sphere is the high pressure accumulator and the post that it's mounted on shouldn't be loose. These accumulators have an internal bladder and these are known to fail with age. Normally these are charged with very high pressure by the system pump.
You'll want to check the large fuse ( 30 Amp ? ) in the brake system fuse block which is on the passenger side inner fender near the air conditioner's receiver/dryer. The fuse supplies the ABS boost motor and these terminals and the attached wiring are known to corrode and fail. You'll want to look at the terminals carefully including the underside.
You'll want to consult the service manual for any of the three years of the TC and in addition, you'll want to reach out and join the only national club that's devoted specifically to the TC's: TC America. You can go to their website for more info:
www.chryslertcbymaseraticlub.com There have been numerous ( more than a dozen ) articles published in the TC America newsletter regarding the care, maintainence and repairs to the Teves system. I'm sure you'll be able to purchase those issues of the newletter that pertain to the system.
The rest of the brake system is based on the components of the 4 wheel disc setup of the 1988 Dodge Daytona ( G body ).