  We introduce and study higher depth quantum modular forms. We construct two
families of examples coming from rank two false theta functions, whose
"companions" in the lower half-plane can be also realized both as double
Eichler integrals and as non-holomorphic theta series having values of "double
error" functions as coefficients. In particular, we prove that the false theta
of $\frak{sl}_3$, appearing in the character of the vertex algebra
$W^0(p)_{A_2}$, can be written as the sum of two depth two quantum modular
forms of positive integral weight.
