Why cannot $\frac{0}{0}$ (indeterminate element) be added to a Field extension to $\mathbb{C}$? Why cannot $\frac{0}{0}$ be added to a field extension to complex field $\Bbb C$? I know the complex field $\Bbb C$ is closed. But can we define additional elements such as $1*\frac{0}{0}$,  $2*\frac{0}{0}$, $i*\frac{0}{0}$ and extend $\Bbb C$ and study its properties? 
 A: Yes, you can add that as an element and get $C\left(\frac{0}{0}\right)$, the field of rational functions over $C$. That is what you obtain by adding an "indeterminate element" to a field.
The question is if you want to require any relations between $\frac{0}{0}$ or its powers and other elements of $C$. 
Some people prefer to call it $x$ if no other relations are required. 
A: If you consider $\frac00$ to be an independent symbol, you can adjoin it to a field. As noted by zambawithkolbasa, this would give the same field as just adjoining an indeterminate, $x$.
On the other hand, if you want $\frac00$ to behave like it looks like it should notationally, you will run into problems:
For instance, in a field, by $\frac{a}{b}$, we mean $a\cdot b^{-1}$, but then $\frac00=0\cdot 0^{-1}$, and this is meaningless since $0$ in a field does not have a multiplicative inverse.
Also, the inverse of $\frac00$ notationally should be the reciprocal, $\frac00$.  But then $\left(\frac00\right) ^2=1$.  However then in your extension field, the polynomial $x^2=1$ has more than two roots ($1, -1, \frac00$).  This is even true in characteristic $2$ where $1$ is a double root.
