ALSO, on an unrelated complex number issue, would a transfinite imaginary or complex number necessarily be a special kind of transfinite number? It seems to me that complex numbers don;t meet the definition of either cardinal or ordinal
-1 isn't even a cardinal or ordinal, so it would be pretty silly to add a square root of -1. If you extend the ordinals to include negatives, subtraction, division, square roots of positive numbers, etc., you end up with the surreal numbers, although the operations end up being somewhat different from cardinal or ordinal arithmetic. You can then add a square root of -1 to get the surcomplex numbers.
Do you surmise it's wise to have laser beams emitting from your eyes?
-They Might Be Giants, "The Lady and the Tiger"