Is an electric bike like this required any special license. The power applied comes from the collected kinetic inertia from going down hill or stored from the rider then the stored power can then be applied while going up hill or to go faster. Are there laws restricting bikes like this anywhere before I buy one?
In my country, a bicycle that is power-assisted with the <=250W <=48V electrical engine, that assist rider up to speed 25km/h is still a bicycle, no matter how energy to power the engine is obtained. All the laws for regular bicycles applies to it - dimensions, equipment. Speed limits apply to such bicycles as to any other kind of vehicle. There are no additional restrictions.
There is no speed limit for a bicycle, but you can be charged for "dangerous use of a vehicle"
An electric bicycle must have no more than 300W of power in addition to the rider's mandatory pedal input. Bicycles do not need registration or licencing, but there are lighting/reflector, braking and helmet requirements.
Any more than 300W and it becomes an electric motorcycle, regardless of whether it has pedals or not. At that point it requires licencing and registration and the rider needs a motorcycle endorsement on their driving license.
If your bike has electric brakes that do energy recovery, then that is irrelevant to the power output of the motor. Your ride does require two independent braking systems, so that failure of one does not prevent the other from working.
By comparison, a liquid fuel motorbike without pedals, and cubic engine capacity of under 50cc still requires registration and licencing, but can be ridden by a licenced rider with a car endorsement only. A motorbike licence is not required.