  We study the fractional maps of complex order, $\alpha_0e^{i r \pi/2}$ for
$0<\alpha_0<1$ and $0\le r<1$ in 1 and 2 dimensions. In two dimensions, we
study H{\'e}non and Lozi map and in $1d$, we study logistic, tent, Gauss,
circle, and Bernoulli maps. The generalization in $2d$ can be done in two
different ways which are not equivalent for fractional-order and lead to
different bifurcation diagrams. We observed that the smooth maps such as
logistic, Gauss, and H{\'e}non maps do not show chaos while discontinuous maps
such as Lozi, Bernoulli, and circle maps show chaos. The tent map is continuous
but not differentiable and it shows chaos as well. In $2d$, we find that the
complex fractional-order maps that show chaos also show multistability. Thus,
it can be inferred that the smooth maps of complex fractional-order tend to
show more regular behavior than the discontinuous or non-differentiable maps.
