Creation of complex numbers

The pair of transforms

$$f(x)=x^2 \qquad f^{-1}(x) = \sqrt{x}$$

applied to real numbers, yields to the complex numbers by the same scheme:

1. Original set: Consider the image of the set of real numbers
   
   $$1, 2, 3, 4, \ldots$$
   
   
   under the transform
   
   $$f(x)=x^2.$$
   
   
   Set of images is positive numbers.
2. Complement: The set of images is naturally complemented to the set $Z$ of all real numbers.
3. Inverse Transform: Now apply the inverse transform
   
   $$f^{-1}(x)=\sqrt{x}$$
   
   
   to the complemented set $Z$ of images.

4. Imaginary Numbers!

   We arrive at new numbers: These are imaginary numbers: The square roots of negative numbers.

Semantic question: Why are the names ``negative,'' ``complex,'' and ``imaginary'' refer to something unpleasant, overcomplicated, or not real?