  Unfolding homoclinic tangencies is the main source of bifurcations in
2-dimensional (real or complex) dynamics. When studying this phenomenon, it is
common to assume that tangencies are quadratic and unfold with positive speed.
Adapting to the complex setting an argument of Takens, we show that any
1-parameter family of 2-dimensional holomorphic diffeomorphisms unfolding an
arbitrary non-persistent homoclinic tangency contains such quadratic
tangencies. Combining this with recent results of Avila-Lyubich-Zhang and
former results in collaboration with Lyubich, this yields the abundance of
robust homoclinic tangencies in the bifurcation locus for complex H\'enon maps.
We also study bifurcations induced by families with persistent tangencies,
which provide another approach to the complex Newhouse phenomenon.
