What your instructor probably had in mind is the following.
Let $z_1=a_1+ib_1$ and $z_2=a_2+ib_2$ be two complex numbers.
Then, the polynomials $P(X)= [\frac{1}{2}X+a_1]+[\frac{1}{2}X+a_2]+i(b_1+b_2)$ has by FTA a complex root, which we can define as $z_1+z_2$.
Similar things can be done for $z_1z_2$ and $\frac{1}{z_1}$.
Anyhow, given a polynomial $P$, even if somehow you can define the polynomial algebraically only by using real operations and $i$, you cannot use the FTA without defining first addition and multiplication. The FTA is not about the polynomial as an algebraic expression, it is about the Polynomial as a function...
To Quote zev, FTA asserts that any non-constant polynomial with complex coefficients has a root in the complex number. But what does a root of a polynomial mean? How do you calculate/evaluate $P(z)$ if you don't know how to calculate the powers of $z$, and multiply $z$ to the coefficients of the polynomial? And how do you add the monomials of the polynomial together?
The only way in which you can get around this issue is by trying to define $C$ as the algebraic closure of $R$, and then the algebraic process of "algebraic closure" and the uniqueness of it shows that any algebraic extension of $R$ in which FTA holds has to be $C$... But then, if you use this definition of $C$, how do you prove that $C= \{ a+bi |a,b \in R\}$??