There is no such thing as an H4 HID bulb. The H4 lamp is a "quartz halogen" dual filament bulb, period.
Some automotive HID headlamp replacement bulbs can be modified to fit into an H4 reflector. The conversion kits you see are using this method to jam an HID lamp in an H4 reflector and provide or modify a ballast to suit. Since the H4 is a dual (hi/lo) beam system you either get one beam HID and one beam still filament, or there's a mechanism to move the bulb a bit changing the focus of the lamp.
The lamps in both H4 and HID headlights are an integral part of the optical system. Swapping them results in a half-assed disaster. Some light comes out, sure. The road may be well lit with an HID lamp in an H4 reflector.
Take a look at an H4 bulb. The "low beam" filament has a metal shield/tiny reflector on one side, this blocks the light that would blind oncoming cars. The HID lamp does not have this shield. Obviously this means that the HID bulb in an H4 reflector will be frikking bright in the eyes on oncoming vehicles... this is the problem.
The DOT regulations are a bit arbitrary in lots of ways, they generally say that non-original safety related equipment is not allowed. Many systems are DOT certified on a vehicle, not just the headlights. Some things, like brake parts, can be certified (it's actually up to a manufacturer to comply with regulations and say a part is certified - then either the DOT or another agency can test it and find it non-compliant BUT the DOT does no testing to certify products for sale) as compliant for certain vehicles. Some things just can not ever be DOT certified compliant... like H4 HID bulbs: the name is itself a contradiction since an H4 lamp must have a tungsten-halogen filament type bulb. This is not just semantic nonsense, the H4 filament size, position relative to the base flange, and shading reflector are all specified within close tolerances. The reflector and lens must take the light from an H4 bulb and focus it to pretty tight luminance and directivity specifications.
There are specifications for the light pattern and brightness minimums and maximums for vehicle headlights in general, they have to put out over a minimum light on the road and shoulders and under a maximum light level in the "glare zone". Someone might be able to design an HID bulb that would meet these specs in an H4 reflector... but this would cost a bunch of $$$ to design and more $$$ to produce... and the bulb would still be impossible to certify because (sorry to repeat) an H4 headlight can only have an H4 tungsten-halogen filament type bulb by definition and by DOT standard. Such a bulb would be, in my opinion, entirely OK to use on my bike just like I have used non-DOT certified braided brake lines because I know they're stronger and safer than the rubber OEM DOT lines and pose no safety problem. But no lamp manufacturer is going to make a "proper" H4 HID lamp because they will necessarily be uncertifiable and can be prohibited from US sale at the whim of the DOT with probably a large fine assessed as well.
Putting a complete DOT legal automotive HID projector headlight unit in a bike will probably still contravene some DOT regulations, but it would - IMO - be OK, as the light pattern would have to be within the standards. I haven't seen HID dual beam units yet but maybe some cars have them... I haven't been looking unto what's on cars I can't afford but there are pretty basic problems designing a projector type lamp with two beam patterns, and putting a secong HID lamp in for high beam is quite costly for the benefit.
The ballast provides high voltage square wave AC with current limiting. There has to be a high voltage ignition system as well. These usually work by switching on the AC power, the lamp will not turn on as the breakdown voltage across the lamp electrodes is well over the voltage supplied. With no current the AC voltage goes abnormally high, and the starting supply is triggered by this high voltage. It provides a DC pulse or a very high AC voltage that sparks across the lamp electrodes. The spark flash is from a small path between the electrodes becoming an plasma - ionized gas that conducts electricity. The AC square wave voltage is plenty high enough to flow through this plasma and it does so, at a much higher current but much lower voltage than the starting voltage. At the lamp's normal operating current the AC supply voltage goes way down. The starting circuit shuts down at this lower voltage. The starters all work the same basic way, the lamp power circuit goes through the secondary coil of what's basically a Tesla coil. The primary is powered either with a very fast switching DC pulse or a traditional RF AC Tesla primary voltage. A simple bypass capacitor shunts the high secondary voltage and protects the power supply - a low value but high voltage capacitor is high impedance at the main switching frequency but is quite low impedance at RF frequencies or for the very fast transient of a DC ignitor. There is a design balance between the switching speed of the main supply and the shunting effect needed to bypass the starting supply.
You need a square wave supple because the lamp is not made for DC, there are DC xenon lamps but these aren't used in automotive lamps and tend to be higher power (500W - 30kW). If a sine wave AC supply was tried the arc would extinguish for too long and you would have to restart it with HV again, the square wave supply does "shut off" the lamp at zero crossings but the gas stays ionized long enough that it will still conduct power when voltage returns. Some lamps like the sodium streetlamps will work on normal 60Hz AC because the gas is chosen for its long lasting ionization - but as far as I know there aren't any true white light lamps possible with these gases.
Oh man... this has become quite a rant. Bye for now!
