A complex path $\gamma$ (path at $\mathbb{C}$), can be defined like a continuous function
$$\gamma: \left[ a,b \right] \subset \mathbb{R} \to \mathbb{C}$$ $$\gamma \left( t \right) = x \left( t \right) + y \left( t \right) i$$
Where $x, ~ y: \left[ a,b \right] \to \mathbb{R}$ are continuous functions. My question is: what is the notation of the set of all paths at $\mathbb{C}$? And how can i define this set, with math symbols?
I don't know of any such standard notation.
Consequently, you can just define a symbol that is aesthetically pleasing or appropriate (and hopefully not overloaded). For instance, if you like denoting individual paths by $\gamma$, I would say, perhaps, $$ \Gamma := \{ \gamma \mid \gamma \text{ is a path in } \mathbb{C} \} $$ Maybe you could have a more convoluted definition, but I can't imagine why anyone would object to this definition -- no need to overcomplicate things, as this object is well-defined (provided "a path in $\mathbb{C}$" is defined).