How Not to Generalize a Derivative: A Complex-Time Difference Quotient That Lands Exactly on the Celerity

We study a nonstandard generalization of the difference quotient for a planar trajectory over complexified time t=α+iβ, obtained by combining the four real directional derivatives a=x_α, b=x_β, c=y_α, d=y_β through an i-weighted harmonic mean. We derive the closed-form modulus\[|v|²={[bc(a+d)]²+[ad(b-c)]²}{(a²+b²)(c²+d²)}\]and show that the construction fails the basic consistency tests expected of a derivative: it does not reduce to dx/dα for purely real motion, and is undefined when $c=d=0$. Surprisingly, under the single condition $d=0$ the modulus collapses to |v|²=a²b²/(a²+b²), independent of c, and its exact inversion yields precisely the relativistic proper velocity (celerity), a=e/√1-(e/b)²=γ u, upon reading e as ordinary velocity and b as the limiting speed. The destination formula is textbook material; the path to it appears to be new. We discuss why this ``wrong'' generalization nonetheless encodes exact relativistic structure on a codimension-one subset of its parameters, and what the example teaches about naive complexifications of kinematics. All symbolic results have been verified with a computer algebra system.