"Real" numbers are mathematical abstractions. Perhaps it's unfortunate that "real" and "imaginary" were used instead of more abstract sounding terms that came with less attached connotation.
The argument of the author is not that the real numbers are poorly named, it's that the mathematical abstraction supposes physically impossible properties of the model. Therefore physics ought to be based on an abstraction that has more physically plausible properties instead of the real numbers.
Not that I necessarily agree with such an argument.
There is value in comparing mathematics to reality even though mathematics is defined entirely independent to reality. We are able to use mathematics to model certain aspects of reality with great effectiveness.... why? Physical reality does not seem to resemble mathematical notions at all, yet shows some of the same results. Understanding that gap would be of monumental significance.
