For what its worth, there are concrete, practical applications for 
binomial coefficients with negative arguments. They are used in 
fractional calculus

https://nrich.maths.org/1365

which in turn has applications in physics, chemistry and other sciences:

https://en.wikipedia.org/wiki/Fractional_calculus#Applications

Take that however you see fit :-)

My own preference is for a comb() function that returns 0 for out of 
bounds values (e.g. "choose 5 items from 4"), and a seperate binomial() 
function that accepts any positive or negative integer values. Even I 
think that fractional and complex arguments are probably a bit too 
exotic for the std lib -- that's Mathematica territory.

And yes, Mathematica does accept fractional and complex arguments to 
the choose() function.

https://www.wolframalpha.com/input/?i=choose%28sqr%28-3.5%29,+sqrt%28-4%2Bi%29%29