Are $mathbb{C}-mathbb{R}$ imaginary numbers?

Question

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?

Eric Towers · Answer

Imaginary numbers are real multiples of $mathrm{i}$.  So the complex number $1+mathrm{i} in Bbb{C} smallsetminus Bbb{R}$ is neither real nor imaginary.

Alekos Robotis · Answer

Depends what you mean by "imaginary." Perhaps you mean an element of $Bbb{C}$ of the form $ai$ for $ain Bbb{R}$ in which case this is false. Indeed, in the complex plane you have removed only the "$x$-axis" so that
$$Bbb{C}setminus Bbb{R}={a+bi:b
ne 0:text{and}:a,bin Bbb{R}}.$$

Are $mathbb{C}-mathbb{R}$ imaginary numbers?

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