Background

I am teaching senior high school students about the structure of numbers.
Start from defining $mathbb{Q}$ and $mathbb{R}$ as the rational and real numbers respectively, we can define $mathbb{R}-mathbb{Q}$ as the irrational numbers.

I am trying to use the same logic to define imaginary numbers by making use of the relationship between $mathbb{R}$ and $mathbb{C}$. Another definition for imaginary numbers is

numbers that become negative under squaring operation.

Let $mathbb{C}$ and $mathbb{R}$ be the complex and real number sets respectively. Are $mathbb{C}-mathbb{R}$ imaginary numbers?