Answer

I didn’t really understand the question. The paradox of the red heifer already appears in the midrash, but it seems to me that there is a logical explanation for it. Someone who deals with the red heifer becomes impure, and then the heifer remains more pure (because it transferred impurity to the priest who is dealing with it), and so now it is prepared and able to purify those who are impure.
But what does that have to do with complex numbers? The conjugate of a complex number is also a complex number (with the imaginary part having the opposite sign). What does it mean to say that the conjugate does the same thing? Numbers don’t perform operations. Perhaps you mean that the operation of taking the conjugate of a complex number performs an action. What it does is reverse the sign of the imaginary part (= a reflection across the real axis). What is paradoxical here?
What physical operation are you looking for? What is it supposed to do that would be similar to taking a conjugate or to the red heifer? I didn’t understand.
——————————————————————————————
Questioner:
A. If there is a logical explanation, that goes against the idea of the midrash. (“I said: I will become wise—but it is far from me…”)
B. I also think this fact should not be pointed to as the reason this is a statute. In complex numbers, the conjugate of 1+i is 1-i, and applying conjugation to 1-i again gives 1+i. By analogy: the ashes render the pure impure, and the ashes purify the impure. The proof from complex numbers is not a good one, because it follows from their definition, so I am looking for a similar idea from the world of physics, on the macro or micro level. If it exists, then there is a problem with the midrash, because this would be natural reality.
——————————————————————————————
Rabbi:
Are you looking for a physical operation that turns X into Y and Y into X?
Lukewarm water cools what is hot and warms what is cold. But that is only a reversal of direction, not a full reversal. Hot water does not actually become cold, but rather something between lukewarm and hot. And similarly with cold things.
But the red heifer also does not perform the two actions simultaneously. It renders the pure impure when it is being prepared, and purifies the impure after it has been prepared. According to the explanation I suggested, that seems to me quite similar to the example of lukewarm water.