Does the set of composable arrows in a category have to be a pullback? 
When defining a category, do we need to have the set of composable arrows be a pullback of the domain and codomain selecting functions? What can go wrong if we use a subobject of the pullback?

This arose when trying to formalize pasting diagrams with a minimal amount of graph theory; I would like to have a category whose objects are complex numbers and whose arrows are injective paths in the complex numbers (and constant paths for identities), to avoid things I don't need when drawing diagrams like loops or space filling curves.
The problem arises because domains and codomains are given by the image of $0$ and $1$ respectively, but composable arrows should have common domain/codomain and not intersect otherwise their composition wouldn't be an injective path and would fail to be in the category. (note that although pasting diagrams are not injective paths since they intersect at some vertices and thusly fail to be arrows in the injective path category, they are still diagrams in this category)
 A: As mentioned in the comments, there are different possible notions of "partial category".  A definition due to Freyd is called a paracategory; this is an "unbiased" notion, equipped with partial $n$-ary composition operations for all $n\ge 0$ (although the case $n=0$ is required to be total, i.e. all identity arrows exist).  This can be reformulated in terms of generalized multicategories, as in the papers by Hermida and Mateus cited at the nLab page.
Another, more "biased" notion of "partial category" is a 2-coskeletal simplicial set with some, but not all, unique inner horn fillers.  Then the existence of a (necessarily unique) 2-simplex can be seen as a witness that a certain composite $g \circ f$ exists and equals $h$.  Again all identities exist, being given by the degeneracies (although one could imagine a semi-simplicial version that might not have all identities), and the composites are biased towards the binary, with the 3-simplices witnessing that associativity holds "whenever it makes sense".
