We extend the definitions of dyadic paraproduct and $t$-Haar multipliers to
dyadic operators that depend on the complexity $(m,n)$, for $m$ and $n$
positive integers. We will use the ideas developed by Nazarov and Volberg to
prove that the weighted $L^2(w)$-norm of a paraproduct with complexity $(m,n)$
associated to a function $b\in BMO$, depends linearly on the
$A_2$-characteristic of the weight $w$, linearly on the $BMO$-norm of $b$, and
polynomially in the complexity. This argument provides a new proof of the
linear bound for the dyadic paraproduct (the one with complexity $(0,0)$).