The map $|N(\cdot)|^{1/n}$ is a continuous multiplicative extension of $|\cdot|$.
By a multiplicative function I mean a function $\chi:L\to [0,\infty)$
such that $\chi(0)=0$, $\chi(1)=1$ and for every $x,y\in L$, $\chi(xy)=\chi(x)\chi(y)$.
A multiplicative function which satisfies for every $x,y\in L$, $\chi(x+y)=\chi(x)+\chi(y)$ is called an abolute value.


**Theorem:** Let $K$ be a field and $|\cdot|$ an absolute value on $K$.
Assume $K$ is complete with respect to the induced metric.
Then for every finite filed extension of $K$, every continuous multiplicative extension of $|\cdot|$ is an absolute value.
In fact, there exists a unique such continuous multiplicative extension of $|\cdot|$, which is $|N(\cdot)|^{1/n}$. 

The uniqueness follows at once from the fact that all norms are equivalent on finite dimensional vector spaces over complete fields: for multiplicative functions $\chi_1,\chi_2$, the function $\chi_1/\chi_2$ (defined on the multiplicative group) is multiplicative too, hence must be unbounded or trivial.
The less trivial part of the theorem is its first part.

In my other answer I gave a proof of this fact which indeed holds in the generality of complete fields. This post is to give an easy proof I found, under the assumption that the fields are locally compact.

We consider a local filed $K$, endowed with an absolute value $|\cdot|$,
a finite filed extension $L$ of $K$ and a continuous multiplicative extension $\chi$ of $|\cdot|$. We argue to show that $\chi$ is an absolute value on $L$.

Consider $L$ as a $K$-vector space and consider the corresponding space $\Omega$ consisting of $K$-norms on $L$.
Consider the multiplicative group $L^*$ as a locally compact group (see Lemma 3 in my other answer for continuity of inversion) and let it act on $\Omega$ by $\|\cdot\|\mapsto \chi(x)^{-1} \|x\cdot\|$
for $x\in L^*$. Note that for $x\in K^*$, 
$$ \chi(x)^{-1} \|x\cdot\|=\chi(x)^{-1} |x|\|\cdot\|=\|\cdot\|,$$
thus the $L^*$-action on $\Omega$ factors via $L^*/K^*$,
which is a compact group, as it is homeomorphic to a projective space.
This action admits a fixed point. Indeed, for every norm $\|\cdot\|\in \Omega$,
the map 
$$ L \ni v \mapsto \int_{L^*/K^*} \chi(x)^{-1}\|xv\|~ \text{dHaar}_{L^*/K^*}(x) \in [0,\infty)$$
is easily seen to be an $L^*$-fixed norm on $L$.
We let $\|\cdot\|$ be such a fixed point which is normalized to satisfy $\|1\|=1$. Then for every $x\in L^*$,
$$ \|x\|=\|x\cdot 1\|=\chi(x)\cdot \chi(x)^{-1}\|x\cdot 1\|=
\chi(x)\cdot \|1\|=\chi(x). $$
Thus $\|\cdot\|=\chi$ and in particular, we conclude that $\chi$ is indeed a norm.