We consider $C^r$ $(r=3,\dots,\infty,\omega)$ diffeomorphisms with a generic
homoclinic tangency to a hyperbolic periodic point, where this point has at
least one complex (non-real) central multiplier and some explicit assumptions
on central multipliers are satisfied so that the dynamics near the homoclinic
tangency is not effectively one-dimensional. We prove that $C^1$-robust
heterodimensional cycles of co-index one appear in any generic two-parameter
$C^r$-unfolding of such a tangency. These heterodimensional cycles also have
$C^1$-robust homoclinic tangencies.