  We discuss an extension of the almost Hadamard matrix formalism, to the case
of complex matrices. Quite surprisingly, the situation here is very different
from the one in the real case, and our conjectural conclusion is that there
should be no such matrices, besides the usual Hadamard ones. We verify this
conjecture in a number of situations, and notably for most of the known
examples of real almost Hadamard matrices, and for some of their complex
extensions. We discuss as well some potential applications of our conjecture,
to the general study of complex Hadamard matrices.
