the oil in the turbo is what suffers from coking, not the turbo itself really
the oil basically burns up in an enclosed space and turns to goo and will not flow
Ok....let me put on my Turbo engineer hat on.
An oil cooled turbo (standard version) can be destroyed by a hot shutdown. Basically what happens is that even though you have shut off the engine, the turbine wheel is still spinning down from 15 - 30k RPM's. The thrust bearing only lives because you are constantly supplying fresh oil (albeit very hot). As soon as you stop the flow of oil, the oil film thickness (at the elevated temp) are insufficient and you can burnish the thrust bearing. In addition, the heat from the turbine side is transmitted through the shaft and will coke (burn off) any remaining oil thats left in the bearings or thrust surface. The coked oil will prevent new oil on the next start up from flowing to all the bearings, you then get metal to metal contact. Not good.
When you do a proper shut-down your doing two things. One, your allowing the turbine to spool down to a low RPM before shutting the engine off. (5k RPM at idle). In addition, the relatively cool exhaust gases at idle draw the heat away from the turbine wheel and shaft and in effect lower the temperature below the threshold of coking. An oil cooled turbo, shut down properly, will live almost indefinitely. This is because all of the bearings are hydrodynamic. Ie. oil film bearings. There are no metal parts physically touching.
A water cooled turbo (water cooled center section) has a separate cavity in the turbo in which you flow engine coolant. This helps to remove heat from the bearings and gives a buffer between the hot turbine side and cool compressor side. (There is a slight performance benefit here.) In addition, during shut down, even a hot shut down, the coolant will flow through the center section and continually cool. Same idea as hot air / cold air. As the coolant is heated by the turbine side, it tends to rise in the cooling system, drawing fresh (colder) coolant in as is flows. A water cooled turbo is MUCH less likely to suffer coking issues from a hot shut down. But its better to be safe than sorry.
NOW.... to answer the original question. There are water cooled turbine housings. (The glowy thing in the picture). They are designed to protect the components AROUND the turbo from the immense heat, and are most common in marine applications inside engine compartments. It is actually detrimental to the performance of the turbo as a whole. As Pilotx1 said, a turbo relies on the hot exhaust gases flowing across the turbine wheel. The turbine wheel takes the heat and pressure and turns it into mechanical energy. If you remove heat from the turbine wheel or housing, you reduce the amount of heat available for spinning.
Instead of water cooling, you want to trap the heat in the turbine housing. Ceramic coatings or even ceramic parts have been used to this effect. In fact I have a ceramic turbine wheel sitting on my desk. Very neat and efficient, but easily destroyed by small high speed particles.