The most chimeric mathematical object I know of is the Moulton plane. Its "points" are ordinary points of the plane $\mathbb{R}^2$, but its "lines" are a chimera, consisting of ordinary lines of non-negative slope, and bent lines of negative slope whose slope doubles as they cross the $y$-axis.
This monster is the standard example of a projective plane in which the Desargues theorem does not hold.
