Are the $p^n$-adic numbers isomorphic to the $p$-adic numbers? In another recent question, the 10-adic numbers came up (along with the usual issues of not really being a field due to 10 not being prime, etc). I had a thought: ordinary binary numbers (either integers or reals) can be trivially interconverted between binary, octal, hexadecimal, etc notations. Is this likewise true of the 2-adic numbers, and if not, why not?
Or, the converse - is there an e.g. "8-adic" ring that is distinct in its properties from the same notation from grouping 2-adic numbers into groups of three digits, etc.?

No, the $8$-adic completion of the rationals is isomorphic to the $2$-adic. You will recognize that if you put $|8|_8=1/8$, then it must follow as the night the day that $|2|_8=1/2$.
