We present a 100 ks Chandra observation studying the extended X-ray emission around the powerful z= 1.04 quasar PKS 1229−021. The diffuse cluster X-ray emission can be traced out to ∼15 arcsec (∼120 kpc) radius and there is a drop in the calculated hardness ratio inside the central 5 arcsec consistent with the presence of a cool core. Radio observations of the quasar show a strong core and a bright, one-sided jet leading to the south-west hotspot and a second hotspot visible on the counter-jet side. Although the wings of the quasar point spread function (PSF) provided a significant contribution to the total X-ray flux at all radii where the extended cluster emission was detected, we were able to accurately subtract the PSF emission using Chandra Ray Tracer and marx simulations. The resulting steep cluster surface brightness profile for PKS 1229−021 appears similar to the profile for the FR II (Fanaroff–Riley class II) radio galaxy 3C 444, which has a similarly rapid surface brightness drop caused by a powerful shock surrounding the radio lobes. Using a model surface brightness profile based on 3C 444, we estimated the total cluster luminosity for PKS 1229−021 to be . We discuss the difficulty of detecting cool-core clusters, which host bright X-ray sources, in high redshift surveys.