The best example I can think of at the moment is Conway's surreal number system, which
combines 2-adic behavior in-the-small with $\infty$-adic behavior in the large. The surreally
simplest element of a subset of the positive (or negative) integers is the one closest to 0
with respect to the Archimedean norm, while the surreally simplest dyadic rational in a
subinterval of (0,1) (or more generally $(n,n+1)$ for any integer $n$) is the one closest to 0
with respect to the 2-adic norm (that is, the one with the smallest denominator).
This chimericity also comes up very concretely in the theory of Hackenbush strings:
the value of a string is gotten by reading the first part of the string as the unary representation
of an integer and the rest of the string as the binary representation of a number between
0 and 1 and adding the two.
I'm having a hard time saying exactly what I mean by chimericity in general, but
some non-examples may convey a better sense of what I don't mean by the term.
A number system consisting of the positive reals and the negative integers would
be chimeric, but since it doesn't arise naturally (as far as I know), it doesn't qualify.
Likewise the continuous map from $\bf{C}$ to $\bf{C}$ that sends $x+iy$ to $x+i|y|$ is chimeric (one does
not expect to see a holomorphic function and a conjugate-holomorphic function stitched
together in this Frankenstein-like fashion), so this would qualify if it ever arose naturally, but I've never seen anything like it.
Non-Euclidean geometries have different behavior in the large and in the small, but the
two behaviors don't seem really incompatible to me (especially since it's possible to
continuously transition between non-zero curvature and zero curvature geometries).
One source of examples of chimeras could be physics, since any successful Theory Of
Everything would have to look like general relativity in the large and quantum theory in
the small, and this divide is notoriously difficult to bridge. But perhaps there are other
mathematical chimeras with a purely mathematical genesis.
See also my companion post Where do surreal numbers come from and what do they mean? .
Best Answer
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.