  Let $K$ be a complex bi-quadratic field with ring of integers
$\mathcal{O}_{K}$. For $K = \mathbb{Q}(\sqrt{-m}$, $\sqrt{n}$), where $ m
\equiv 3 \pmod 4 $ and $ n \equiv 1 \pmod 4$, we prove that every algebraic
integer can be written as sum of integral squares. Using this, we prove that
for any complex bi-quadratic field $K$, every element of $4\mathcal{O}_K$ can
be written as sum of five integral squares. In addition, we show that the
Pythagoras number of ring of integers of any CM field is at most five.
Moreover, we give two classes of complex bi-quadratic fields for which
$p(\mathcal{O}_{K})= 3$ and $p(4\mathcal{O}_{K})=3 $ respectively. Here,
$p(\mathcal{O}_{K})$ is the Pythagoras number of ring of integers of $K$ and
$p(4\mathcal{O}_{K})$ is the smallest positive integer $t$ such that every
element of $4\mathcal{O}_{K}$ can be written as sum of $t$ integral squares.
