A pedantic question about notation -- "such that" symbols

In modern mathematical papers $\ni$ is almost exclusively used to mean contains, as in, "the integers contain $3$" would be written as $\mathbb Z \ni 3$. If you intend that others read your mathematics then I would highly recommend you stick to using $:$ or $|$ for "such that" in set builder notation. If your notation is just for your own use then it doesn't really matter what you use as long as you know what it means.

The backwards $\in$ you are talking about isn't actually a backwards $\in$. It is backwards epsilon. The usage was introduced by Peano specifically to mean "such that".

The only modern usage I've ever really seen is more like a comma (small, shallow concave-left curve, lowered with respect to the line of text) with a dash through it. It doesn't look like a backwards $\in$ at all. Certainly, Peano's backwards epsilons look like backwards epsilons. But not much like the modern, highly stylized epsilon $\in$ or $\ni$.

The problem I have with $\ni$ is that it makes an optical illusion in combination with $\in$: $$ \forall z \in \mathbb{R} \ \exists w \in \mathbb{C} \ni w^2 = z $$ My eye is drawn to the $\mathbb{C}$ and I can't focus on the formula. Also there's the issue that $A \ni x$ is used as syntactic sugar for $x \in A$, and I find this more useful.

In written work I would use no symbol for “such that”—just the words. In notes I will use “s.t.” which works fine. $$ \forall z \in \mathbb{R} \ \exists w \in \mathbb{C} \text{ s.t. } w^2 = z $$ When logicians use $\exists$, the “such that” is implicit. So you could technically use no symbol, but I don't know how readable that is.
$$ \forall z \in \mathbb{R} \ \exists w \in \mathbb{C} (w^2 = z) $$ Actually I think they write it more like this: $$ \forall z (z\in\mathbb{R} \rightarrow\exists w(w\in\mathbb{C} \wedge w^2=z)) $$