Why exactly are complexity theorists interested in closed timelike curves? Context:

There are several papers that study the implications of closed timelike curves (CTCs) to quantum complexity. In 2008, Aaronson and Watrous published their famous paper on this topic which shows that certain forms of time travel can make classical and quantum computing equivalent i.e. quantum computers provide no computational advantage if they can send information to the past through closed-timelike curves. 

Questions:


The abstract clearly says that closed timelike curves are not known to exist. So why exactly are complexity theorists interested in this topic? Does the study of CTCs provide some non-trivial insight into the fundamentals of complexity theory? 
Are there any other world lines that are popularly studied in the context of complexity theory? If yes, why? If not, why not (and then what's so special about CTCs)?


I haven't really gotten around to working through the CTC papers, but I'm trying to get an idea of the "big picture" here, so as to understand the motivation behind studying this topic. 



Note: I previously asked about this on Quantum Computing SE, in the context of quantum information theory, but here I'm specifically trying to view it through the lenses of a complexity theorist or computer scientist.
Sorry for the very "big picture" answer from a non-quantum-theorist, but this contrast might help: you could describe algorithms as the study of how to solve computational problems, whereas complexity theory studies the resources (especially time) required in theory to solve them. How hard are these problems really at some fundamental level, and how are they classified by resource requirements? This question is considered interesting regardless of how many or what kinds of resources puny humans happen to have at their disposal.

(For instance, an $n^{100000}$ time algorithm for SAT would totally upend our view of complexity theory even if it never impacted any algorithms we could run in this universe. A $2^{0.00000001n}$ time algorithm for SAT would change our understanding of complexity much less, even though it would have huge practical implications.)

The question of how much time is needed in theory to solve a problem changes if you allow CTCs, therefore they are interesting to complexity theory by definition.

So I feel your first question is like asking why computability theorists would study Turing degrees higher than zero; it's what they do.