This is how to answer the question (but it's not an answer).

Let $K_1$ and $K_2$ be the fields in question. By local class field theory you're asking if $\mathcal{O}_{K_1}^\times\times\widehat{\mathbb{Z}}$ and $\mathcal{O}_{K_2}^\times\times\widehat{\mathbb{Z}}$ are isomorphic (the usual exact sequence splits). Because everything is finitely-generated an equivalent question is whether $\mathcal{O}_{K_1}^\times$ and $\mathcal{O}_{K_2}^\times$ are isomorphic. Now for $K=K_1$ or $K_2$ we have $\mathcal{O}_K^\times$ is a finitely-generated group isomorphic in this case to the product of a finite group of odd order (which you can compute by computing the odd order torsion, which is the same as the torsion in the residue field by a Hensel argument) and the Sylow 2-subgroup of $\mathcal{O}_K^\times$, which is the kernel $1+m_K$ of the reduction map onto the residue field. Finally $1+m_K$ is a finitely-generated $\mathbb{Z}_2$-module so is classified by its torsion subgroup and rank.

For $K=K_1$, $K_2$ both degrees are 8 ($x^8-3$ and $x^8-48$ are both irreducible over $\mathbb{Q}_2$ according to pari) so all you have to do is to check that the torsion in the unit groups are the same and this looks like a much easier question. However I don't know how to do this offhand without more thought. This question should be easy with a computer though -- compute the residue fields and then try and fathom out for which $n$ you have a $2^n$th root of unity in each $K_i$.

This is how to answer the question (but it's not an answer). (Edit: it and the comments below now form an answer).

Let $K_1$ and $K_2$ be the fields in question. By local class field theory you're asking if $\mathcal{O}_{K_1}^\times\times\widehat{\mathbb{Z}}$ and $\mathcal{O}_{K_2}^\times\times\widehat{\mathbb{Z}}$ are isomorphic (the usual exact sequence splits). Because everything is finitely-generated an equivalent question is whether $\mathcal{O}_{K_1}^\times$ and $\mathcal{O}_{K_2}^\times$ are isomorphic. Now for $K=K_1$ or $K_2$ we have $\mathcal{O}_K^\times$ is a finitely-generated group isomorphic in this case to the product of a finite group of odd order (which you can compute by computing the odd order torsion, which is the same as the torsion in the residue field by a Hensel argument) and the Sylow 2-subgroup of $\mathcal{O}_K^\times$, which is the kernel $1+m_K$ of the reduction map onto the residue field. Finally $1+m_K$ is a finitely-generated $\mathbb{Z}_2$-module so is classified by its torsion subgroup and rank.

For $K=K_1$, $K_2$ both degrees are 8 ($x^8-3$ and $x^8-48$ are both irreducible over $\mathbb{Q}_2$ according to pari) so all you have to do is to check that the torsion in the unit groups are the same and this looks like a much easier question. However I don't know how to do this offhand without more thought. This question should be easy with a computer though -- compute the residue fields and then try and fathom out for which $n$ you have a $2^n$th root of unity in each $K_i$.

This is how to answer the question (but it's not an answer).

Let $K_1$ and $K_2$ be the fields in question. By local class field theory you're asking if $K_1^\times$ and $K_2^\times$ are isomorphic. Now for $K=K_1$ or $K_2$ we have $K^\times\cong\mathbb{Z}\times\mathcal{O}_K^\times$, and $\mathcal{O}_K^\times$ is a finitely-generated group isomorphic in this case to the product of a finite group of odd order (which you can compute by computing the odd order torsion, which is the same as the torsion in the residue field by a Hensel argument) and the Sylow 2-subgroup of $\mathcal{O}_K^\times$, which is the kernel $1+m_K$ of the reduction map onto the residue field. Finally $1+m_K$ is a finitely-generated $\mathbb{Z}_2$-module so is classified by its torsion subgroup and rank.

For $K=K_1$, $K_2$ both degrees are 8 ($x^8-3$ and $x^8-48$ are both irreducible over $\mathbb{Q}_2$ according to pari) so all you have to do is to check that the torsion in the unit groups are the same and this looks like a much easier question. However I don't know how to do this offhand without more thought. This question should be easy with a computer though -- compute the residue fields and then try and fathom out for which $n$ you have a $2^n$th root of unity in each $K_i$.

This is how to answer the question (but it's not an answer).

Let $K_1$ and $K_2$ be the fields in question. By local class field theory you're asking if $\mathcal{O}_{K_1}^\times\times\widehat{\mathbb{Z}}$ and $\mathcal{O}_{K_2}^\times\times\widehat{\mathbb{Z}}$ are isomorphic (the usual exact sequence splits). Because everything is finitely-generated an equivalent question is whether $\mathcal{O}_{K_1}^\times$ and $\mathcal{O}_{K_2}^\times$ are isomorphic. Now for $K=K_1$ or $K_2$ we have $\mathcal{O}_K^\times$ is a finitely-generated group isomorphic in this case to the product of a finite group of odd order (which you can compute by computing the odd order torsion, which is the same as the torsion in the residue field by a Hensel argument) and the Sylow 2-subgroup of $\mathcal{O}_K^\times$, which is the kernel $1+m_K$ of the reduction map onto the residue field. Finally $1+m_K$ is a finitely-generated $\mathbb{Z}_2$-module so is classified by its torsion subgroup and rank.

For $K=K_1$, $K_2$ both degrees are 8 ($x^8-3$ and $x^8-48$ are both irreducible over $\mathbb{Q}_2$ according to pari) so all you have to do is to check that the torsion in the unit groups are the same and this looks like a much easier question. However I don't know how to do this offhand without more thought. This question should be easy with a computer though -- compute the residue fields and then try and fathom out for which $n$ you have a $2^n$th root of unity in each $K_i$.