What is the symbol for imaginary numbers?

The imaginary numbers on their own are boring. I'll denote the imaginary numbers by $\mathbb{I}$. Let's see how it's structured.

Firstly, we can definitely add an subtract them and you will end up with an imaginary number. But can we multiply them? If you multiply two non-zero imaginary numbers you get a non-zero real number, so it's no longer in $\mathbb{I}$.

How about other properties? Well you can define a total order on $\mathbb{I}$ by using the ordering on $\mathbb{R}$. You can also show that it's complete. But $\mathbb{R}$ is the unique totally ordered field, and $\mathbb{I}$ has been shown to be "inferior" to $\mathbb{R}$ in that it's not closed under multiplication.

This shows that the "set of imaginary numbers" isn't a useful concept. That's why you've probably never seen a label for the set. However if I were to choose one symbol it would be $i\mathbb{R}$.

tl;dr $i\mathbb{R}$, but it's a pointless concept.

There is an accepted symbol for complex numbers, $\textbf{C}$. The others seem to have assumed you already know this symbol, but from your question it is not entirely clear if you do or do not.

If you are referring to purely imaginary numbers, numbers $z$ such that $\Re(z) = 0$, then the answer is no, there is no accepted symbol for imaginary numbers.

The consensus here is that if you really need a symbol, you could go with $\textbf{I}$ but would be much better off with $i \textbf{R}$.

And yeah, it is easy to conclude that the purely imaginary numbers are "boring." After all, they are closed under addition but not multiplication, whereas purely real numbers are. This is also true of real rational numbers.

But we should also consider the context in which you need a symbol for purely imaginary numbers. Do you mean to say that a particular number $bi$ is purely imaginary? You could write $bi \in i \textbf{R}$. But then it would be just much easier to write $b \in \textbf{R}$, and then it is clear that $\Re(bi) = 0$.

It seems a little strange that $0$ is both purely real and purely imaginary. If you need to say that $bi$ is a nonzero purely real imaginary number, you could write that $\Re(bi) = 0$ but $\Im(bi) \neq 0$.

It's important to realize that although there are purely imaginary numbers of the form $ai$ (where $a \in \mathbb R$), that these numbers are only a subset of Complex numbers.

The symbols for Complex Numbers of the form $a + bi$ where $a, b \in \mathbb R$ the symbol is $\mathbb C$.

There is no universal symbol for the purely imaginary numbers. Many would consider $\mathbb I$ or $i\mathbb R$ acceptable. I would.

Note:

$\mathbb R = \{a + 0*i\} \subsetneq \mathbb C$. (The real numbers are a proper subset of the complex numbers.)

$i\mathbb R=\{0 + b*i\} \subsetneq\mathbb C$. (The purely imaginary numbers are a proper subset of the complex numbers.)

$\mathbb C = \{a+b*i\} \subseteq \mathbb C$.

[Fun Trivia Fact: $0$ is an imaginary number. $0$ is a real number. $0$ is the only real number that is imaginary and the only imaginary number that is real.]

Also notice: Despite all the fan fare about learning that Imaginary Numbers exist, they actually aren't in the least bit interesting or important. We use them to define the Complex Numbers which are important (and interesting) but the set of purely imaginary numbers is really only a sidestep on the way to a result.

There isn't, and the problem is that the most "natural" choice, $\mathbb I$, is already overloaded. The Mathworld page on doublestruck symbols gives only one meaning for $\mathbb I$: integers. The OEIS Wiki page on the Latin alphabet gives imaginary numbers as the primary meaning of $\mathbb I$, without citation, then gives "integers, more commonly $\mathbb Z$" as a second meaning and gives the Mathworld page as a citation for that.

So that leaves $i \mathbb R$, the second choice most people suggest, as the more viable alternative. This is a symbol that should feel familiar and comfortable to anyone who has studied principal ideals even cursorily. After all, what is a purely imaginary number but a real number multiplied by $i$? Even $i$ itself can be thought of as $1 \times i$.

As we're dealing with commutative algebra (right?), $\mathbb R i$ is an acceptable variant, a distinction without difference.