>he origin goes back to Cardano solving cubic equations.

Whether or not mathematicians realized this at the time, there is no functional difference in assuming some imaginary number that when multiplied with another imaginary number gives a negative number, and essentially moving in more than 1 dimension on the number line.

Because it was the same way with negative numbers. By creating the "space" of negative numbers allows you do operations like 3-5+6 which has an answer in positive numbers, but if you are restricted to positive only, you can't compute that.

In the same way like I mentioned, Quaternions allow movement through 4 dimentions to arrive at a solution that is not possible to achieve with operations in 3 when you have gimbal lock.

So my argument is that complex numbers are fundamental to this, and any field or topological construction on that is secondary.