Alpha City is an impressive structure, and is a ziggurat arcology sitting on artificial island surrounded by an artificial moat/lake. Twenty-six major roads connect Alpha City to it's supporting satellite districts. The lower tier of the ziggurat is just over half a mile tall, and has a large open platform that functions as an aerodrome and base for aerostats and a functional tethervator (space elevator precursor that uses aerostats and magnetic tethers to move containers ships higher and higher until it can be boosted into orbit. The rest of the ziggurat surface serves as the ground floor for over a dozen super highrises, creating an utterly unique skyline for the city. The 'steps' of the ziggurat have more super highrises in rows, creating the image of stairs leading to heaven.
The island is reinforced material and serves as the base of a massive Castle Avalon. Many of the super highrises are shells that house the hundred meter long barrels of anti-ship gauss cannons that hit targets in orbit. There are concealed missile silos housing building size missiles that can reach targets in lunar orbit. The core of the Castle Avalon is the Alpha City geofront, which is the great and massive arsenal of the city. It contains dozens of polyforge factories, munitions factories, and body farms for churning out weapons and soldiers to use them.
The moat is an entirely artificial body of water and is does double duty as a heat circulation system for the industries inside the geofront, and as a host for a tropical aquarium style seacology. People live in the lap of luxury under the water, looking at massive genimal fish, supersized versions of salt water coral reef fish.
Places of Interest:
Olympus - the ultra-luxury demi-arco that is the central feature of the skyline, and the base of the tethervator, Olympus is among the tallest structures on Earth, and it houses the hyper and ultra wealthy who make their residence in the city. Most of these individuals have multiple residences, often including homes on other planets, in orbit, and even private islands or space habitats entirely to themselves.
Grand Mezzanine - the Grand Mezzanine is a massive commercial and common space district that is the core of Alpha City, and is located in the center of the ziggurat. It has sunlight directed into the core, and is as much a commercial and cultural focal point as it is a literal hanging garden. The entire sector is a display of technological prowess and wizardry, leaving many thinking that the area is many magnitudes larger than it actually is.
Castle Ultima - the official title of the Castles Avalon structure, Ultima is an armored curtain wall wrapped around the geofront, and houses not just the mecha barracks, aerodrome, and the more than impressive anti-ship weaponry concealed within.
The Empty City - despite all of the splendor and technology in Alpha City, the city is empty. No one is interested in placing anything built as such a massive fortress. Having a billionaire's wealth means that you don't have to live in or anywhere near something as strategic as a mega-fortress. The city isn't truly empty, it still has staff and residents, but these are people who have been assigned to the city for various reasons. Employees of service industries, but with no customers. One version of this rumor is that Alpha City is a grandiose real estate venture that went tits up. Another version is that the city is a honeypot, a lure to draw in potential enemies so they can be revealed and dealt with. The most popular version is that Alpha City was a headhunter trap, and that the powerful realized that if they ever stepped foot in the city, it would become their prison, and they would be made into the puppets of those in charge of the city.
The Satrapy Configuration - the problem of Alpha City is that is not self sufficient as many arcologies are. It is instead reliant on the territories around it for food, water recycling, and so forth. The workforce required to support Alpha City would number in the tens of thousands, and would only grow as the city population increased. This is expanded to cover that Alpha city has no green towers, no internal manufacturing ability.
An Accident - the construction of Alpha City has almost no record or paper trail, and to most of the world it rather sort of just appeared. The rumor is that the existing information was falsified and that Alpha City literally did just appear out of nowhere, a massive fortress city, devoid of inhabitants, but bristling with weaponry. Most shrug, dimensional fatigue event? and move on with their lives.