Computers operate not in Base 16 (Hex) but Base 2 (Binary). Hex is only used as a convenient way for us humans to handle binary.
Any ways to have an architecture with a different base than base16?
I don't know any operating in base 16. Today essentially all computers operate base 2. In the olde times (tm) there were many base 10 and a few base 3 machines.
The only partitial system operating in base 16 is the floating point format used by IBM's /360 mainframe family. The /360 follow up systems are kind of an oddity about that (*1). But then again, this is only how the values are handeled. In storage they are again binary. The only other base still supported by many CPU is decimal (base 10), but as well only for operation, as storage is as well binary (4 bit per decimal number).
I was reading this Wikipedia entry [about numeral systems] and thought if any other Computer/or Architecture was ever made with a different base system than base16.
Beside, there was never one, the Wiky entry also doesn't claim such to ever existed. It only stats that Base 16 (aka Hexa/Sedecimal) is used as "compact notation for binary data". Much the same way as it states for octal.
Thinking of it, do you maybe mixup presentation of binary data to a computer user with the number system used?
Base 8 or base 16 are, much the same way as base 10, used to make binary readable to average users. It's simply handy for us to operate with a 4 digit hex number instead of 16 binary digits - isn't it?
Which number system us used or human representation depends not on the CPU, but convention. Octal comes traditionally from classic machines with word sizes being a multiple of 3, while hex is originated in IBM's /360 series, as it was the first machine to widely use multiples of 4 as word size (BCD/Nibble, Byte, Halfword, Word, Doubleword)
And if not, any ways to do it? (maybe by making one from scratch or reprogramming a specific Cpu?)
As said, (today's) computers do not operate in base 16, but base 2. Their data is only displayed to us humans as base 16. But yes, a computer can be made operating on units in any base you want. Except, it will make the machine more complicated and more expensive.
Binary is not only most simple system, as there is no other with less elements, it als fits quite well the way electronics work. Systems to a power of two (base 4/8/16/32/...) would fit as well, with only a minimal overhead, but bring no advantage to base 2. Any other system will end in much more complex hardware and lower performance.
But why doing it at all? The data can be converted anyway. So the most sensible step is to use the most basic format, and convert it later to anything the user wants.
(might be related to this question)
It's not really a good idea to repeat questions with essentially the same content that have been rejected before.
*1 - Amdahl's (he headed that development) decision was based on the fact that the /360 ISA was already build around a datapath to handle 4 bit values for BCD - which was very important back then and still is today for accounting - so it doing hex based float did not only come at a lower price tag than binary, but also benefited from all improvement made to sped up BCD.