Panasonic has been talking about an organic sensor that may have some interesting properties for a few years now, but (as far as I'm aware) we haven't seen even a working prototype.
The very best current sensors seem to be bumping up against the limits of conventional sensor design (if we define conventional as 14 bit Bayer (or X-Trsns, which is a variant of Bayer) sensors using standard materials).
The low-hanging fruit may be getting more dynamic range by moving to a 16-bit readout while improving noise/photosite capacity farther. The best current sensors have meaningful information in all 14 bits, so more information would need more readout bits. A few medium format sensors (and a RED sensor or two?) already use 16-bit readouts.
You can't go too much farther with resolution before getting bitten both by diffraction and reduced photosite capacity (a 60ish MP 24x36mm sensor should work - both APS-C and medium format sensors are at that density and working well). The 20 mp Micro 4/3 sensor is a comparatively weak performer - perhaps because it's too dense (that would be about 36 MP APS-C, 80 MP 24x36mm).
Organic materials may offer more photosite capacity, allowing more DR at the same photosite sizes, (needs 16-bit readout), smaller photosites without sacrificing DR (watch out for diffraction), or both.
Multi-layer sensors could offer interesting improvements, especially in color rendering. So far, Bayer and X-Trans sensors have been able to beat any multi-layer sensor for most images, simply by offering enough more resolution and DR that they capture most or all of the color information that the multi-layer sensor gets while also getting much more luminance and spatial resolution. The latest multi-layer sensor that has been announced is Sigma's L-mount 20 MPx3 design (next year). It will offer 20 MP each of red, green and blue - but only 20 MP of spatial resolution. It will probably lag the best of the competition in DR - Foveons always have. A 45 MP Bayer sensor actually offers more green resolution than the Foveon (22+ MP), which the eye is most sensitive to, while lagging in red and blue. However, it offers more than twice the spatial resolution and a couple extra stops of DR. A 45 MPx3 Foveon would almost certainly outperform 45 MP Bayer (substantially), especially if the DR was the same. However, the Foveon process has never been at a stage where it offers similar spatisl resolutions.