You can do electromagnetism with fields other than the electric and magnetic fields and get the same result. In fact, if you're a moving observer, electric fields will start to generate magnetic fields and vice-versa.

In reality, electric and magnetic fields are underlying manifestations of the same (coordinate-independent) underlying "thing". This thing, electromagnetism as a unified object, is more properly described by the four-potential, of which the electric and magnetic fields are parts. We call the part that doesn't care whether you're moving "electric" and the part that does "magnetic". But that is only a choice of coordinates, in the same way that you can do your linear algebra with <1,1> and <1,2> as basis vectors rather than <1,0> and <0,1> and everything will work out fine.

It turns out that you can, in a sense, decompose any (non-changing) vector field this way into an "electric-like" field and a "magnetic-like field", in the sense that these fields share some of the important mathematical properties of the electric and magnetic fields respectively.