Can a non-simple graph have a complement? If so, this means that:
Two different graphs can share the same complement, however each graph cannot have two different complements.
Is this correct?
 A: The notion of complement of a graph is usually restricted to simple graphs. For example, please see the Wikipedia definition. (Go to the beginning of the "Formal construction" part.) 
One could extend the definition to graphs that allow a single loop from a vertex $v$ to itself. In that case, the answer to your question would be no. With multiple loops or edges, there is no nice definition of complement.  For one thing, a desirable property of a "complement," shared by other things called complement, is that the complement of the complement of $X$ is $X$. 
A: Well, if the set of all possible edges can be well defined and fixed, then --relative to it-- one can define the complement, and it will be still unique (for example if $e,f$ are possible parallel edges, and $e\in\mathcal G$ but $f\notin\mathcal G$, then $f$ will be in the complement of $\mathcal G$).
A: complement of a simple graph G=(V,E) is a graph having the same set of vertex but it has a complement edge of G. In multiple graph the complement does not defined, also in a graph that not multiple but has a self loops the complement of it is defined by adding a self loop of every vertex that does not have the self loop.  One example of component graph G is show below.
A: yes,  A complement of a simple graph G=(V,E) is a graph having the same set of vertex but it has a complement edge of G. In multigraph ( i.e graph that has a parallel edge) the complement does not defined, also in a graph that not multigraph but has self loops the complement of it is defined by adding a self loop of every vertex that does not have the self loop.
