I think, oil quenching will provide higher hardness due to rapid cooling. In general duplex stainless steel is water quenched immediately after solution annealing.
The faster the cooling rate the higher the hardness (in steels). Water-quenched steels will generally be harder than oil-quenched steels. This is mainly because the thermal conductivity of water is higher than the thermal conductivity of most oils (that I know); consequently, the rates of cooling will be less rapid (or lower) in oils compared with water. Also, see TTT and CCT phase diagrams for more information (if you haven't done so already).
I agree with Dr. Uyime, water quenching is faster than oil quenching. However, apart from thermal conductivity, other factors affecting the heat transfer (convective) are listed below:
1. Specific heat of water is higher than most regular oils.
2. Lower viscosity of water compared to oils.
3. Density of water is higher than that of oils.
All these above result in water having a higher convection coefficient as compared to oils (for removal of heat from the hot material) and thus water has a better quench efficiency.
Faster the cooling, greater is the suppression of the diffusion dynamics of carbon in the crystal lattice and higher the c/a ratio of the BCT unit cell of Martensite (post quenching), thus improving the hardness.
The severity of quenching for water medium is more than that of oil. What matters in duplex stainless steel is sequence and holding time at particular temperature. For that TTT curves should be consulted for a particular grade.
To my literature background, water quenching is recommended for all DSS grades for several reasons, however, oil quenching medium is not ignored it is also considered a good one and less vulnerable to precipitations.