  In this paper, we define and study a variant of multiple zeta values (MZVs)
of level four, called alternating multiple mixed values or alternating multiple
$M$-values (AMMVs), which forms a subspace of the space of colored MZVs of
level four as ${\mathbb Q}[i]$-vector spaces. This variant includes the
alternating version of Hoffman's multiple $t$-values, Kaneko-Tsumura's multiple
$T$-values, and the multiple $S$-values studied by the authors previously as
special cases. We exhibit nice properties similar to the ordinary MZVs such as
the duality, integral shuffle and series stuffle relations. After setting up
the algebraic framework we derive the regularized double shuffle relations of
the AMMVs. We also investigate several alternating multiple $T$- and $S$-values
by establishing some explicit relations of integrals involving arctangent
function. In the end, we discuss the explicit evaluations of a kind of AMMVs at
depth three and compute the dimensions of a few interesting subspaces of AMMVs
for weight less than 7.
