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Is there really any meaningful sense in which real and imaginary numbers have the same units but two dimensionless real hyperparameters don't?

Complex numbers multiplied by an imaginary number rotate preserving magnitude. In this sense they have the same units.

You can rotate any vector while preserving its magnitude.

If the units are different, you do need to come up with a conversation between them, or you're implicitly saying it's 1-to-1