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Elementary algebra may explains to readers certain core complex number ideas quite well, but I only put it out there for discussion.  Argand diagrams made complex numbers respectable.  These diagrams are metaphors, far from exact ones, of the imaginary field, yet appear in the article. Imaginary numbers are not yet points.  Keith Devlin wrote ""since the complex numbers are not points on a line, you cannot say which of two complex numbers is the larger; for the complex numbers there is no such notion."" Yet he also stated ""There is, however, a notion of size of sorts.  The absolute value of a complex number a + bi is the distance from the origin to the number."" [Devlin, Keith, The Language of Mathematics: Making the Invisible Visible Macmillan, 2000: 133.} Desmond Fearnley-Sander explains that ""From a geometric point of viuew, complex numbers are not points but ratios of vectors."" [Fearnley-Sander, Desmond, Plane Euclidean Reasoning, in Automated Deduction in Geometry, 2nd International Workshop, ed. Gao, X, et. al,  Beijing, China 1998, Google eBook: 108] They're the clock hand that points to the hour, not the hour point. 

Consistent with this logic, it's better to consider the plane rotating under the complex vector than the vector itself.  This sidesteps the variability of complex number position: its the plane's fault, if you will.  Want this grounded in mathematical foundations?  Look no further than the Riemann hypothesis.

Riemann extended Euler's zeta function, which Euler developed to find prime numbers, to define complex numbers.  Riemann's wanted to prove Euler's conjecture, which he attacked by solving complex solutions to f(zeta) = 0.  Riemann found x + iy, where x lies between 0 and 1, has infinitely many solutions.  He proposed they all (except negative even integers) are of form 1/2 + iy.  Computer tests find the first 1.5 billion f(z) = 0 (called z(s)) solutions lie on the line x = 1/2 in the complex plane.  But inductive evidence does not mathematically prove anything, and the Riemann hypothesis remains perhaps the most important unsolved problem of our era.

The only singularity of z(s) is the simple pole at s, or 1. There is no reason to assume complex numbers are points.  I don't believe this is non-standard. It is profound and documented.
Brian Coyle"