I've highlighted the important part below. Your car does have special dampers. Changing them out for BC coilovers (which are very much a budget coilover) would be crazy in my opinion.
I think the only coilover worth changing to for such a special car would be the AST coilovers that Litchfield put in their Type 25
Rev. F (2005)
The WRX STi receives the same increase in wheelbase as the Spec C. All STis received additional rear arch flaring in order to allow an increase in wheel size. The wheel size changed from 17" x 7.5" to 17" x 8". In accordance with the increased wheel width, steering lock-to-lock turns have been reduced. Wheel hub strength was improved. The P.C.D. was changed from 5 x 100 mm to 5 x 114.3 mm. The interior was vastly improved as well. A whole new center console now brings out a very high grade interior feel.
The WRX STI continues on relatively the same. The Impreza line's looks have been updated.
The WRX STI Spec C's receives the Arai tuned dampers and a 21 mm (0.8 in) swaybar. The rear suspension is changed from rubber bushings to pillowball bushings. The strut tower was further stiffened for improved handling and stability. The strut towers were reinforced for better handling and better ride characteristics. The engine mounts were changed from metal to liquid-filled plastic mounts to reduce vibrations into the car's cabin. Some 2006 owners have had issues with these plastic engine mounts. The manufacturer will replace them, under warranty if broken, with the 2005 rubber mounts.[6]
The DCCD is updated with the use of a torque-sensing mechanism inside the DCCD. The torque split is changed to 41/59.
The WRX STI Spec C was used for Group N homologation