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In Australia, you currently need a recreational helicopter pilot's license to fly this machine – but US owners might not. "It is possible to fly the Pegasus without a pilot's license," reads the company website, "if you have successfully completed the FAA private Pilot written examination and have also completed the company-mandated vehicle familiarisation and operator training programs."
Although somewhat tangential to this community's intent, I decided to verify this claim, since licensing and registration of an air-and-land machine is likely to be a top question.
Regarding United States aviation, everything revolves around certificates. For individuals, a certificate is a license to pilot or operate particular types of aircraft. For machines, a certificate is the authorization of airworthiness for a particular type of aircraft, allowing it to be used in US airspace. Setting aside the road registration quandary -- which is regulated on a state-by-state level -- I will focus only on the aviation aspects, since those are controlled mostly by federal law.
For pilot licensing, there are a fair number of those, ranging from the Student Certificate to get into the air under instruction, to the traditional Private Pilot Certificate/License (PPL) for flying solo or cross country, to the Air Transport Certificate to fly paying customers commercially.
Given the description from the company, I would guess that they meant the Light Sport Certificate, which is an abbreviated pilot license to fly smaller aircraft with up to one passenger and weighing less than 1320 pounds (600 kg). This certificate does not require the thorough medical exam of a PPL, if the holder also has a US Driving License. This might sound a bit weird, since why would driving an automobile be indicative of sufficient health to not crash an airplane, but it's a balancing act given the restrictive set of aircraft types that can be flown with a Light Sport certificate.
So the company's statement is vaguely, mostly correct, if they meant that the strict requirements of a PPL can be avoided, by instead applying for a Light Sport certificate. Although this still requires 20 hours of flight training beforehand.
As for certificating the aircraft itself, this is fairly straightforward, since the manufacturer just needs to declare that they meet all the requirements in 14 CFR § 1.1. The FAA would then grant the type certificate, allowing this aircraft into the national airspace. The owner would then need to register the aircraft by presenting the type certificate, and then receive a tail number (aka N registration) to attach to the aircraft.
So if a person is in possession of a valid Sport Light certificate, an N registration with the FAA, and this helicopter/go-kart, they should be good to take off, right? Well, mostly, but with a substantial number of caveats.
Firstly, certificated aircraft -- even for the Light Sport or Experimental categories -- is still strict, and any modifications to the airplane that deviates from the regulations can invalidate the type certificate. Even just basic maintenance must be performed by a certificated mechanic, as well as the mandatory annual inspection. While one could obtain that additional certificate to maintain one's own Light Sport aircraft, the inspection must take place at a certificated inspection location, which is probably somewhere else.
And then there are the operational limitations. This is less of a problem with the machine itself, since the limits in 14 CFR § 1.1 simply won't impose a restriction. Rotorcraft don't usually fly high enough that they need pressurization, and this machine only comes in a one- and two-seat variant. The real issue is the pilot's license limitations.
With only the Light Sport pilot license, it is Visual Flight Rules (VFR) only. So no flying at night, through or over the clouds, nor into inclement weather where visibility is below 3 miles (5 km). And only Class E and G airspace are permitted, unless having obtained additional endorsements to operate near airports and to communicate using the radio.
This page describes US airspace circa 1997 for ultralights (a category of very light flying machines that the FAA doesn't even require type certification for), but it does underscore the complexity in determining what airspace is of which class. Ultralights and LSA do tend to use the same airspace, though.
Perhaps you could argue this go-kart/helicopter is only meant for flying close to the surface, so the roughly 700 ft upper bound for Class G airspace would be perfectly sufficient. Maybe. But most urban areas have an airport within 10-15 miles (16-25 km), which may displace the Class G airspace to a 0 ft upper bound, meaning the Class G space is non-existent. That really cuts against using this as an urban commuter.
But maybe it's meant for rural/suburban commuters who don't necessarily go into the urban core but travel around it. But now we've reduced this machine to a one-trick pony.
Do I think this is an intriguing machine? Yes, absolutely; I wouldn't have hashed out this long comment if it weren't at least some food for thought. But do I think this is the future for the masses? Definitely not.
The process to get this legally flying is sufficiently involved that it will never see mass appeal. And I will not even entertain the notion that flying regulations should be relaxed to accommodate this novelty, since we already have an example where lax licensing of heavy machines and grossly insufficient operator training causes thousands of deaths per year needlessly: automobiles.