The discussion in these posts, Completions of number fields and Corresponding local extension does not depend on choice of prime ideal above?, gets at the fact that if $L/K$ is a Galois extension of number fields and $P,Q$ are primes of $L$ both above a prime $p$ in $K$, then there is an isomorphism $L_P \cong L_Q$ as local field extensions of $K_p$.
When $L/K$ is not Galois, this can fail to be true. But in all the example I have seen so far, the primes $P,Q$ above $p$ have different ramification/residue field indexes.
Question: If $L/K$ is a non-Galois extension of number fields and $P,Q$ are prime ideals of $L$, both above a prime ideal $p$ of K, which have the same residue degree and ramification index, is it the case $L_Q \cong L_P$ as fields over $K_p$?
For example, when $L/K$ is not Galois, $[L:K]=n$, and $p O_L = P_1 P_2 ... P_n$ (i.e. $p$ is totally split in $L$), are the extensions $L_{P_i}/K_p$ ($i=1,...,n$) all isomorphic as field extensions of $K_p$?
Edit: I think in this second, $p$ totally split in $L$, case $[L_{P_i}:K_p] = e(P_i/p)f(P_i/p)=1$, so all the $L_{P_i}$s are $K_p$.