Imaginary numbers are more real than transcendental "real" numbers, and much more real than the unnameable reals, which are almost every real. It's easy to set up situation that points to any algebraic complex number.
Is there a definition you have in mind for "unnameable"? If you mean definable [0][1], then this is independent of ZFC.
The "standard" argument for this is flawed, and indeed there are models where every real is definable. [2]
[0] "x is definable if there exists a first order formula with one free variable P such that x is the unique real number with P(x) true" [1] https://en.wikipedia.org/wiki/Definable_real_number#Definabi... [2] https://mathoverflow.net/questions/44102/is-the-analysis-as-...