What Re(f(z))=c is if f is a holomorphic function? Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function. 
If $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then it follows from implicit function theorem that 
$\text{Re}(f(z))^{-1}(c)$ is at least locally a differentiable curve in the plane. 
 Question: 
1- If $c$ is a regular value is any connected component of $\text{Re}(f(z))^{-1}(c)$ a global differentiable curve ?
2-If $c$ is not a regular value and $\text{Re}(f(z))^{-1}(c)$ have at least one cluster point is this set locally a curve ?

(1) To use your notation, if $c \in \mathbb{R}$ is a regular value, then the level set $Re(f(z))^{-1}(c)$ will be a 1-dimensional embedded submanifold of $U \subset \mathbb{R}^2$.  Therefore, every connected component will be a connected 1-manifold.
Now, I'm not sure what exactly you mean by "global differentiable curve."
If you mean "something of the form $f(t) = (x(t), y(t))$," then it follows from the comments in this question that yes, every connected component can be put in that form.
If you mean "something of the form $y = f(x)$ or $x = f(y)$" then the answer is (I think) no.  For example, consider $f(z) = \log z$ on $U = R^2 - \{x \geq 0, y = 0\}$, i.e. the plane with the non-negative x-axis deleted.  Then $Re(f(z)) = \log(\sqrt{x^2+y^2})$, so the level set $Re(f(z))^{-1}(1)$ is the circle $x^2 + y^2 = e^2$ minus the point $(e,0)$.  So, it doesn't seem like you'd be able to represent this curve by a single function $x = f(y)$ or $y = f(x)$.
