  The previous constructions of quadrature amplitude modulation (QAM) Golay
complementary sequences (GCSs) were generalized as $4^q $-QAM GCSs of length
$2^{m}$ by Li \textsl{et al.} (the generalized cases I-III for $q\ge 2$) in
2010 and Liu \textsl{et al.} (the generalized cases IV-V for $q\ge 3$) in 2013
respectively. Those sequences are presented as the combination of the
quaternary standard GCSs and compatible offsets. By providing new compatible
offsets based on the factorization of the integer $q$, we proposed two new
constructions of $4^q $-QAM GCSs, which have the generalized cases I-V as
special cases. The numbers of the proposed GCSs (including the generalized
cases IV-V) are equal to the product of the number of the quaternary standard
GCSs and the number of the compatible offsets. For $q=q_{1}\times q_{2}\times
\dots\times q_{t}$ ($q_k>1$), the number of new offsets in our first
construction is lower bounded by a polynomial of $m$ with degree $t$, while the
numbers of offsets in the generalized cases I-III and IV-V are a linear
polynomial of $m$ and a quadratic polynomial of $m$, respectively. In
particular, the numbers of new offsets in our first construction is seven times
more than that in the generalized cases IV-V for $q=4$. We also show that the
numbers of new offsets in our two constructions is lower bounded by a cubic
polynomial of $m$ for $q=6$. Moreover, our proof implies that all the mentioned
GCSs over QAM in this paper can be regarded as projections of Golay
complementary arrays of size $2\times2\times\cdots\times2$.
