The Base Power

A technological fortress built into an island volcano. A castle on a cloud. A mysterious tower that only exists at night. A space station accessed by a teleportation pad. An aircraft carrier that flies. What do these have in common?

They all make no logical sense as bases of operations (why would anyone spend that much to build it?), and yet show up regularly in comics for villians and heros. They also tend not to outlast their associated primary character, but be back in businesses not long after the primary character returns. Why?

They're a form of super power. There's no logical sense to how they work or why they were built because nobody really built them and they don't use standard physics. They break down shortly after the demise of their owners because the power that created and maintained them stops flowing when their owners perish. They are created by simply forming around where certain people spend most of their time. They don't make logical sense, they make thematic sense, matching their owner's powers and personality because they're ultimately a manifestation from the owner's subconscious.