What is the mathematical motivation for complexifying momenta in BCFW? One of the first steps in obtaining the on-shell BCFW recursion relations is complexifying the momenta of the external particles. Now complexifying things is not unprecedented (the dispersion program showed us how poles represent bound states), but here it, well, happens so much you can’t help but be annoyed.
Is there some mathematical reason for allowing complex momenta? (Like the ${}^{\mathbf C}\mathfrak{so}(3,1) = \mathfrak{so}(4,\mathbf C)$ complexification that physicists love to ignore in QFT courses.) Or at least is there some before-the-fact motivation to do this, something that does not boil down to “look, I just pretended they were complex, and a bunch of cool complex analysis stuff popped up and solved my problem”?
Reference: Elvang, Huang, “Scattering amplitudes”, arXiv:1308.1697 [hep-th], section 3.
This is a rather vague question, but it turns out that even some people known for their work in amplitude techniques have nothing to say to this! I have to entertain the possibility that we just don’t know yet, and I’d love to be proven wrong.
 A: "A bunch of cool complex analysis stuff popped up and solved my problem" is about as honest as it gets. But physicists do this from more or less their first differential equation: using $e^{i \omega t}$ to track both solutions via the cool-ness of complex analysis. There's no a priori or manifestly physics-based reason to do it that way.
In the "original" BCFW paper (actually the second one of BCF with Witten where they proved the relations), they have some discussion of this in very read-able format. They suggest thinking of is as analytic continuation of the variables of the function - in this case external momenta. Analytic continuation shows up all over the place in physics, in a way that mathematicians are more-or-less okay with. (Oddly Elvang and Huang don't seem to mention this interpretation.) This is maybe not motivation, but it's at least in line with arguments most mathematicians might not find tenuous.
To more directly address your question: one possible post hoc argument comes from the amplituhedron story. There you start with some geometric definition of a polytope, compute its volume in some prescribed way, and the volume so computed is exactly the integrand of the scattering amplitude (in planar $\mathcal{N}=4$ sYM). Moreover, it also happens that the various possible triangulations for computing the volume of the polytope exactly correspond to different choices of BCFW shifts. So this geometric construction of the amplitude appears to "want" the momenta to start life as complex numbers. Britto, Cachazo, and Feng likely had no inkling of this when they wrote the first paper. And I don't know that it counts as "better" motivation; it's just a different point of view.
Maybe with ten more years of research we'll have a better understanding of what's actually going on.
Update: Per the comments, it might be possible to take a very mathematics-oriented approach to learning about the amplituhedron and the detailed chain of logic that connects it to complex shifts in momentum.
The first amplituhedron paper here is self-contained in the sense that they start with a definition of the amplituhedron and go from there.
The second paper here gets into how the analytic structure dictated by the geometry relates to well-studied physical properties of the amplitude.
Here is the most pedagogical paper on the connection between pure mathematics and BCFW shifts. Although the amplituhedron hadn't yet been born when they wrote that, many of the tools that went into the amplituhedron are fleshed out. It's also a book now!
The caveat to all of this is that this technology only currently exists in a limit of a simple theory (planar $\mathcal{N} =4$ super-Yang-Mills). How and if this all works in more general QFTs is ongoing research.
