That's an interesting thought, especially with the addendum of the no-cloning theorem.

This fits with a Hofstadter-like perspective that consciousness is about "strange loops", where we somehow repeatedly evaluate simplified models of the world including yourself. Doing such repeated evaluations requires (at least partial) "cloning" of the state of the world outside for the purpose of evaluation, and cloning quantum states is impossible, hence consciousness must be a classical phenomenon.

I like that line of argument.

Note: I wouldn't state it as being unable to perceive things that can be described as complex numbers, but rather complex linear combinations.

Then wouldn't an appropriately constructed sensor be able to record the full linear combination of quantum states?

No, because records are necessarily classical. This is because of the no-cloning theorem. You cannot copy quantum information, only classical information.