We introduce and study the Hyers–Ulam stability (HUS) of a Cayley quantum
(q-difference) equation of first order, where the constant coefficient is allowed to range over
the complex numbers. In particular, if this coefficient is non-zero, then the quantum equation
has Hyers–Ulam stability for certain values of the Cayley parameter, and we establish the
best (minimal) HUS constant in terms of the coefficient only, independent of q and the
Cayley parameter. If the Cayley parameter equals one half, then there is no Hyers–Ulam
stability for any coefficient value in the complex plane.
