A 2-D vector field also has the properties required to be an algebraically defined number. I don't recall them all; operators with identities and inverse operators, existence of zero, etc. That is why so many concepts in analysis can be treated as a complex number or as a 2-D vector. In electronics, for example, the amplitude and phase of a wave, voltage, impedance, etc. are all expressed as a complex number in analysis when they could also be understood as a vector. It is really about choosing a notation that facilitates stealing analysis techniques from another perspective in the world of math.