It isn't nonsense at all. Subspaces, fields, and vector spaces are all interrelated. The complex numbers are a vector space and the quadratics are a subspace of them. I was hoping that instead of just complex numbers, you could use the special type for all sorts of vector math, tensors, quadratics, values with units attached. Why make complex numbers the special case? Why not have two tuples or n-tuples that know about co-efficients so you can do all kinds of interesting algebra in the same way you can use complex numbers.
All you have to do to make this really really flexible is allow users to change the value of that sqrt(-1) co-efficient(s) and change the length of the vector/sum/value. Tada.
UPDATE: Yes, well, I believe the quadratics are a 2-dimensional vector space and a field of reals. It happens to be a subspace of the complexes, so maybe it doesn't matter, but I fail to see why there couldn't be an interface to use the complex number system for more.