After just having learned about $p$-adic numbers I've now got another question which I can't figure out from the Wikipedia page.

As far as I understand, the $p$-adic numbers are basically completing the rational numbers in the same way the real numbers do, except with a different notion of distance where differences in more-significant digits correspond to small distances, instead of differences in less-significant digits. So if I understand correctly, the $p$-adic numbers contain the rational numbers, but not the irrational numbers, while the non-rational $p$-adic numbers are not in $\mathbb{R}$ (someone please correct me if I'm wrong).

Now the real numbers do not depend on the base you write the numbers in. However the construction of the $p$-adic numbers seems to depend on the $p$ chosen. On the other hand I am sure that the construction of the real numbers can be written in a way that it apparently depends on the base, so the appearance might be misleading.

Therefore my question: Are the $p$-adic numbers the same for each $p$ (that is, are e.g. $2$-adic and $3$-adic numbers the same numbers, only written in different bases), or are they different (except for the rational numbers, of course). For example, take the $2$-adic number $x := ...1000001000010001001011$ (i.e. $\sum_{n=0}^\infty 2^{n(n+1)/2}$), which IIUC isn't rational (because it's not periodic). Can $x$ also be written as $3$-adic number, or is there no $3$-adic number corresponding to this series?

In case they are different, is there some larger field which contains all $p$-adic numbers for arbitrary $p$?