In an elementary context, $\Bbb W$ means the set of whole numbers. Some books have it as
$$\Bbb W=\{0,1,2,\ldots\}$$
while others have it as
$$\Bbb W=\{1,2,\ldots\}$$
Because of the ambiguity, I recommend that you avoid the use of $\Bbb W$. For the second meaning use $\Bbb Z^+$. There still is no perfect abbreviation for the first. Either meaning is also called the Natural numbers, although usually the Whole numbers are meant to be different from the Natural numbers. Again, we see the ambiguity: even if the Whole and Natural numbers are different, which is which? See the Wikipedia articles for a variety of notations for these sets, most of which are far from perfect.
I have occasionally seen the phrase "whole numbers" used for the Integers, which includes negative numbers such as $-1,-2,\ldots$ and is usually written $\Bbb Z$. But I have never seen the notation $\Bbb W$ used in that way.
As @RenatoFaraone points out, in an advanced context $W(x)$ probably means the Lambert W function. But I have never seen that written in the "blackboard bold" style that $\Bbb W$ uses.
The hard truth, as the comments show, is that there is no standard choice of a single symbol for denoting the composite numbers. There aren't very many single symbols available, so any time a person adopts a new single symbol to stand for something, there is a huge risk of ambiguity.
There are many ways of dealing with this risk in mathematical writing.
One is that mathematicians, as a whole, are not easily won over to new notations with one symbol when a simple notation of 3 symbols might do. For example, if I grant you (which I cast doubt on in my comment) that $\mathbb{P} \subset \mathbb{N}$ is a good and standard notation for the prime numbers as a subset of the natural numbers, then the notation $\mathbb{N}-\mathbb{P}$ or $\mathbb{N} \setminus \mathbb{P}$ for the composite numbers is a perfectly clear, it communicates instantly what is meant, and it is only two symbols more than being just one symbol. What's not to like about that?
Well, maybe you are writing a paper in which you need a symbol for the composite numbers, and you need to use it a lot, and you don't want to clutter up the paper with three symbols over and over and over when one symbol can be chosen (I myself find this unconvincing, because 3-to-1 is such a tiny ratio, but then I imagine a 42-to-1 situation and I am more convinced). Or, as your more recent edit suggests, what you really want is to use the symbol as a subscript (this is more convincing for a 3 symbol notation, although I've seen subscripts like that sometimes). In this situation, you are free to pick your own 1 letter symbol to stand for the composite numbers, even if it clashes with standard notation. But there's a catch.
For example, everybody loves $x$. There are even jokes about $x$. And there is a significant proportion of mathematical papers, or proofs within papers, in which the 1-letter symbol $x$ is used to stand for something special. Often $x$ stands for a real number, but just as often it stands for something else. So there is a gigantic risk of ambiguity when one uses the symbol $x$. The way we deal with this in our writing is that we state, clearly and carefully to avoid all possible ambiguity, what $x$ stands for in the limited context of our own paper or our own proof.
So, in whatever you might be writing, you should feel free to state, carefully, clearly, what $\mathbb{C}$ stands for in the limited context of what you are writing. If what you are writing uses both the composite numbers and the complex numbers, well then you have an ambiguity problem to solve, and perhaps you really do not want to use $\mathbb{C}$ for the composite numbers because yes, it is standard for the complex numbers.
But if what you are writing has nothing at all about complex numbers, well then, you should feel free to use $\mathbb{C}$ for the composite numbers, as long as you state clearly and carefully and without ambiguity what $\mathbb{C}$ stands for in your paper. And if somebody screams at you for abusing $\mathbb{C}$, you may want to pick something else. Just be clear and careful about it.
Best Answer
The unambiguous notations are: for the positive-real numbers $$ \mathbb{R}_{>0} = \left\{ x \in \mathbb{R} \mid x > 0 \right\} \,, $$ and for the non-negative-real numbers $$ \mathbb{R}_{\geq 0} = \left\{ x \in \mathbb{R} \mid x \geq 0 \right\} \,. $$ Notations such as $\mathbb{R}_{+}$ or $\mathbb{R}^{+}$ are non-standard and should be avoided, becuase it is not clear whether zero is included. Furthermore, the subscripted version has the advantage, that $n$-dimensional spaces can be properly expressed. For example, $\mathbb{R}_{>0}^{3}$ denotes the positive-real three-space, which would read $\mathbb{R}^{+,3}$ in non-standard notation.
Addendum:
In Algebra one may come across the symbol $\mathbb{R}^\ast$, which refers to the multiplicative units of the field $\big( \mathbb{R}, +, \cdot \big)$. Since all real numbers except $0$ are multiplicative units, we have $$ \mathbb{R}^\ast = \mathbb{R}_{\neq 0} = \left\{ x \in \mathbb{R} \mid x \neq 0 \right\} \,. $$ But caution! The positive-real numbers can also form a field, $\big( \mathbb{R}_{>0}, \cdot, \star \big)$, with the operation $x \star y = \mathrm{e}^{ \ln(x) \cdot \ln(y) }$ for all $x,y \in \mathbb{R}_{>0}$. Here, all positive-real numbers except $1$ are the "multiplicative" units, and thus $$ \mathbb{R}_{>0}^\ast = \mathbb{R}_{\neq 1} = \left\{ x \in \mathbb{R}_{>0} \mid x \neq 1 \right\} \,. $$