Is Redis Turing Complete? Is Redis with all its data types (Strings, Lists, Hashes, Sets, Sorted Sets, etc.) Turing Complete?
No and in a sense yes.

No - this question is strongly reminiscent of those about SQL and the conclusion that it isn't (Turing complete). Despite its many capabilities, Redis does not provide looping or recursion for via native API, so that pretty much nails it.

However, you can easily run Lua scripts in Redis and Lua is in fact Turing complete, so in a sense so is Redis (similarly to how SQL becomes so once you allow the use of PL/SQL or the likes).