Let's consider the following quaternion:
$$ Q=a+b.i+c.j+d.k = \left[ \begin{matrix} a && b && c && d \end{matrix} \right]$$
The normalization is given by the following formula:
$$ Q_{normalized}=\frac{Q}{| Q | } $$
where:
$$ | Q | = \sqrt{ Q.\overline{Q} } = \sqrt{ \overline{Q}.Q }=\sqrt {a^2 + b^2 + c^2 + d^2} $$
The general representation is given by:
$$ Q_{normalized}= \left[ \begin{matrix} \frac{a}{\sqrt {a^2 + b^2 + c^2 + d^2}} && \frac{b}{\sqrt {a^2 + b^2 + c^2 + d^2}} && \frac{c}{\sqrt {a^2 + b^2 + c^2 + d^2}} && \frac{d}{\sqrt {a^2 + b^2 + c^2 + d^2}} \end{matrix} \right] $$