Specifically for the mountain-biking crowd who could use the extra traction, or for people who bike a lot in the snow.
Instead of driving a chain, the pedals would drive a variable displacement hydraulic pump. Displacement (swash plate angle) would be controlled via wire by a control similar to a standard bicycle gear selector - only it would be on the frame and not the handlebars, and be infinitely variable. From the pump, the oil would pass through a flow divider and from there to motors on the front and rear axles. Oil flow and return to the front wheel motor would pass through hydraulic slip rings in the fork hinge, allowing complete 360 degree rotation of the front forks. As well, by means of a bypass valve (a second control on the frame), oil return could be used to charge an accumulator. I see this as being potentially useful when cycling downhill, as you could charge the system while coasting downhill, and then use that energy on demand for a power assist up the next incline. Brakes would not be in the form of pads (i.e. physical friction on the wheel), but rather as a variable flow restrictor in the oil return - controlled by brake handles as per a normal bicycle. By not having the brake pads, you gain clearance around the wheel/tire, allowing the use of tire chains for traction in the mud and snow.
The problems are:
Friction. I would have to use a low viscosity oil which maintains its performance when cold, and I would have to size all of the oil lines appropriately to keep things moving without undue resistance. This worsens the second problem:
Weight. Obviously, you still have to keep the bike as light as possible. I don’t know if you can get off-the-shelf hydraulic components which are lightweight aluminium - it may require custom machining. In any case, even with the lightweight components, this will still be significantly heavier than a standard mountain bike. That, I guess, is the price to be paid for two wheel drive and one of the coolest looking (even if it doesn’t perform to expectations) bikes around.
Think it’ll fly? (No, not the bike - the project…)