Find the roots of the simple equation?

Question

x^{2}= 0 What are the roots? are they in complex plane, but how? Answer seems trivial in real numbers ain't it? Does this evolve a new system like it was with iota?

Answer 1

The root is $0$, multiplicity two. It is a real root, but real numbers are also complex numbers. Any real number $x$ can be expressed as $x + 0i$.

$$z = 0 + 0i = 0 \in \mathbb C$$

No need to evolve a new system to be able to express $0$.

Answer 2

You should observe that $\mathbb{C}$ is a field. Hence, if $ab=0$ then either $a$ or $b$ is zero.

In other rings containing $\mathbb{R}$ you could have more solutions, but not in a field.