Suppose $\alpha \in \overline {\Bbb Q}^*$ be an algebraic number.
For $\alpha$ we define ${\mathrm{N}}(\alpha) \colon= constant~ term ~of ~the~ monic~ minimal~ polynomial~f(x) ~ over~ {\Bbb Q}~such~that~f(\alpha)=0$.
Then is this multiplicative? That is, does the equality ${\mathrm{N}}(\alpha\beta) = {\mathrm{N}}(\alpha) {\mathrm{N}}(\beta)$ hold?
No. The norm is multiplicative if you restrict to a fixed number field and define it in the usual way, but the issue with your definition is that the field generated by $\alpha \beta$ may be different from the fields generated by $\alpha$ and $\beta$.
E.g., take $\alpha = \beta = \sqrt 2$. Then with your definition
$$N(2) = 2 \ne (-2)(-2) = N(\sqrt 2) N(\sqrt 2).$$