Conformal map but not analytic Is there an example of a conformal map(which preserves angles with orientation) which is not analytic defined on any open subset of complex plane?
 A: No. The two concepts are equivalent for functions whose derivative never vanishes.
A: Conformality (and orientation preservation) will allow you to construct a complex derivative (as in "linear approximation") at each point, and complex differentiable functions are holomorphic, which implies analytic. Not that any of what I've said here is trivial (and there might be caveats that I'm forgetting), but it's how I thought your question is most naturally answered.
A: You say "which preserves angles with orientation": it would be better to say "which preserves angles with orientation between any two curves" or equivalently "whose differential preserves angle with orientation, not necessarily the map itself"). Among other thing this implies that conformal maps must be at least $\mathbb{R}$-differentiable.
"Complex analytic" in an open set means $\mathbb{R}-$differentiable in that open set and with the differential being representable multiplicatively by a complex scalar, that is, the differential being a $\mathbb{C}$-linear application, that is, the differential being a $\mathbb{R}$-linear application that transforms the tangent vector at any point of every curve into a vector whose length and angle are incremented w.r.t. to the tangent vector by a non-negative factor and an additive quantity respectively that are dependent only by the point and not by the tangent vector. 
"Conformal" means $\mathbb{R}$-differentiable and such that the oriented angle between the vectors into which its differential transforms the tangent vectors of any two curves at their intersection point is the same of those tangent vectors. 
You can now see that begin complex analytic implies being conformal only if in the above "complex analytic" description non-negative is specialized in positive (that means non-null differential), else you cound not even measure the angle if one or both of the two transformed vectors has a zero length.
But it is also that conformal implies complex analytic, because if the factor of the length increment operated by an operator on a tangent vector depends on the tangent vector itself, while at the same time the additional amount for the angle does not depend on the tanget vector, then this operator is not a $\mathbb{R}$-linear operator and consequently not a differential.
