MIBBLLII
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MIBBLLII Isn't brainfuck But Looks Like It Is is a programming language made by Nathan van Doorn. It is based on the wikipedia:B,C,K,W system. It maps input (expressed as a church list of church booleans corresponding to the bits of the input, ending with an infinite stream of church trues) to output (expressed as a similar list, presumably finite).
Combinators
Expression  Evaluates as 

+ a b c 
a [ b c ]

 a b c 
a c b

. a b 
a

, a b 
a b b

> a 
a

< a b c 
a c [ b c ]

[
and ]
are used to group combinators for evaluation order.
Proof of Turing Completeness
SKI combinator calculus is Turingcomplete, and all the operators in SKI exist in MIBBLLII as <
, .
, and >
. It is still Turingcomplete without <
or >
, as the following shows.
< +[+,][++] > ,.