Emergent complexity and Turing Completeness
I suggest you  to give  a look to S. Wolfram's book "A New Kind Of Science" (that is also available online). 

The centeral ideas are:


Computation is everywhere
Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication (Principle of Computational Equivalence)
Computation irreducibility: while many computations admit shortcuts that allow them to be performed more rapidly, others cannot be sped up.


Though these ideas are probably not new, they are exposed in a fascinating way (to be honest, a rather informal and non-mathematical "way").