I am new to the concept of partially ordered sets.
Here's my professor's definition of a Möbius function from her lecture notes:
The inverse of zeta function with relative to the convolution product is called the Möbius function and denoted by $\mu$.
That is $\mu : \text{Int}(P) \to \mathbb{C}$ is the unique function that satisfies
$Z * \mu = \mu * Z = \delta$
I have also realized that the co-domains of other functions with domain $Int(P)$, set of intervals of $P$, have co-domains of $\mathbb{C}$. Why isn't it any other set?
Eventually you'll be using these functions to do linear algebra to your posets. For instance, for finite posets we can write the zeta function $\zeta$ as a certain matrix, and then the möbius function $\mu$ is just the inverse matrix. This lets us use the power of linear algebra to solve certain problems in combinatorics. Of course, if we want our linear algebra to be nice, we want to be working over $\mathbb{C}$ (or at least an algebraically closed field of characteristic 0) since this is where we have access to as many theorems as possible. For instance, over $\mathbb{C}$ every matrix has an eigenvalue, whereas this is not necessarily true over other fields (like $\mathbb{R}$).
I hope this helps ^_^