video of the talk  

Soergel bimodules have natural manifestations in 3 different contexts: combinatorial (i.e. diagrammatic calculus), geometric (i.e. perverse sheaves on the flag variety) and representation theoretic (i.e. Harish-Chandra bimodules/category O).  

In each of these contexts, there are generalizations that might interest you:

• on the combinatorial side, there is a categorification of the kth tensor power of C^n via KLRW algebras, studied by Khovanov-Lauda-Sussan-Yonezawa in the context of "categorical symmetric Howe duality"; they propose that the action of S_k on this tensor power categories to an action of Soergel bimodules, and prove this for n=2.
• on the geometric side, you can replace a flag by a sequence of maps V_1 -> V_2 -> ... -> V_m -> C^n (considered up to isomorphism) without requiring injectivity.  You can convolve B-equivariant sheaves on this space X with perverse sheaves on B\G/B. 
• on the representation theoretic side, you can replace category O by Gelfand-Tsetlin modules (the modules over U(gl_n) which are locally finite under the Gelfand-Tsetlin subalgebra S).  Like category O, these carry an action of translation functors, which are effectively a copy of Soergel bimodules.

In fact, all of these generalizations are the same! The modules over appropriate KLRW algebras are a graded lift of the category of Gelfand-Tsetlin modules, and Koszul dual to the category of B-equivariant perverse sheaves on X, and all of these equivalences are compatible with the actions of Soergel bimodules.  I'll try to explain this result, and how whatever your perspective, none of the objects involved are as scary as you might think.