  Nonlinear, i.e., multi-photon microscopy is a powerful technique for imaging
deep into biological tissues. Its penetration depth can be increased further
using adaptive optics. In this work, we present a fast, feedback-based
adaptive-optics algorithm, termed C-DASH, for multi-photon imaging through
multiply-scattering media. C-DASH utilises complex-valued light shaping (i.e.,
joint shaping of amplitude and phase), which offers several advantages over
phase-only techniques: it converges faster, it delivers higher image quality
enhancement, it shows a robust performance largely insensitive to the axial
position of the correction plane, and it has a higher ability to cope with, on
the one hand, scattering media that vary over time and, on the other hand,
scattering media that are partially absorbing. Furthermore, our method is
practically self-aligning. We also present a simple way to implement C-DASH
using a single reflection off a phase-only spatial light modulator. We provide
a thorough characterisation of our method, presenting results of numerical
simulations as well as two-photon excited fluorescence imaging experiments.
