Apparently, the C42 nose pitches down shortly after stall, like in this video. I know part of the video might be pilot input to recover from the stall, but I've read on various flying forums that this will happen naturally without input anyway. Why is this?
The Ikarus C42 is very similar to the Cessna 152 in design and power (but around 500 lbs lighter!). This is a classic high wing design that is one of the safest aircraft one can fly. Rock stable on all three axes and very forgiving. These are a lot of fun to slip, as you will not see a lot of roll coupling with yaw (rudder) input.
As far as the video, this is a fairly normal stall recovery. As with anything, practice makes perfect. Properly weighted, this type of plane will not need a hugely aggressive push forward for stall recovery. Just hold the nose straight with the rudder.
There are really 2 lines of defense in stall recovery with an aircraft properly balanced. The first is a natural nose down with loss of speed caused by positioning CG ahead of center of lift from wing and trimming with downforce on tail (elevator up). Because the trim torque is aerodynamic, it has less force when the plane slows down and the nose drops. Planes set up this way will gently drop out of stall simply by relaxing the elevator.
The second line of defense comes from the surface area of the horizontal stabilizer (as seen from underneath the plane). If the plane starts sinking in a nose up or level position, vertical motion will help flip the nose down sharply. This is assisted with down elevator (stick forward).
The key is to hold the nose straight with the rudder. Do not use the ailerons. With the Ikarus, simply letting go of the stick should be enough to recover, but it would be highly recommended to take a lesson and learn the limitations and behavior of that plane.
Most aeroplanes are designed with the Centre of Gravity being ahead of Centre of Lift, so when the aeroplane's wing cannot produce sufficient lift anymore -- due to a high Angle of Attack, the nose will drop, therefore, it will decrease the AoA and as the gravity pulls the aeroplane towards earth, the speed will increase which in combination with the lower AoA will hopefully produce enough lift to keep the aeroplane flying again.
In some cases when the CG is not forward of CoL -- e.g. mostly in transport category aeroplanes, the horizontal stabiliser is set/installed on the neutral nose-down Angle of Incidence, which makes the aeroplane fly in a more stable manner and will help the stall recovery by pushing to nose-down.
Generally flying in any type of aeroplane with CG aft of CoL is considered a no-no, for the reasons explained above: 1) Flight instability and 2) possibly unrecoverable from a stall -- which in fact is likely to happen because of the displaced CG will tend to increase the AoA which will decrease the speed, until stall happens.
Most (if not all) longitudinally stable aircraft will pitch nose down after a stall.
This is because the forward flying surface (regardless of conventional or canard layout) -- or forward portion of the wing, in the case of tailless designs -- must fly at a higher loading and coefficient of lift than the rear in order to maintain stability, so when lift is lost, it will be lost first at the higher-loaded and higher-coefficient surface, which will then start to drop before the lower-loaded or lower-coefficient surface.
As a stall is approached, the center of lift moves slowly forward. The pilot will compensate for this with either back-pressure on the stick/yoke or by trimming the aircraft. When the stall occurs the center of lift suddenly and rapidly moves towards the rear of the aircraft. Since the aircraft was balanced before, moving the force of lift backwards will cause the rear of the aircraft to rotate upwards, and the nose to drop. In a traditional aircraft configuration, the wing will stall before the horizontal stabilizer. The horizontal stabilizer normally produces lift in the downwards direction just incase it does stall, that way the nose would automatically drop. Canard configurations achieve the same effect by having the horizontal stabilizer produce positive lift and stall before the wing.