I'm sorry, but the authors of this paper are totally clueless.

"whenever simple cells start to become more complex"

The definition of complexity here is irrigorous to the point of laughability. At one point the definition makes a call about "size" - " if a bacterium grew to the size of a complex cell, it would run out of juice"

Nonsense. There was recently discovered a macroscopic bacterium - http://en.wikipedia.org/wiki/Thiomargarita_namibiensis Yes, you can see them with your own eyes.

"If the energy-producing machinery straddling the membrane is not constantly fine-tuned, it produces highly reactive molecules that can destroy cells. Yet fine-tuning a larger membrane is problematic because detecting and fixing problems takes longer."

The solution, then is to create short-length response pathways. Presumably the "reactive molecule" they're talking about is superoxide - Life has evolved an incredibly awesome enzyme that takes care of it - superoxide dismutase - and even a slightly defective SOD doesn't lead to inviability. Certain types of photosynthetic bacteria, which need to "carefully manage reactivity" have solved this problem by creating 'hydrogenases' which bleed off excess energy in the form of hydrogen molecules (these may become useful in the alternative fuel industry).

"Acquiring mitochondria, it seems, was a one-off event."

Well, no. Plants later acquired chloroplasts. Some organisms have hydrogenosomes (which is inetresting, because we now have discovered vertebrates that seem to be able to survive off their hydrogenosomes, mitochondria be damned). Some bacteria have evolved anammoxasomes.

"cells capable of complex functions – such as communicating with each other and having specialised jobs – could evolve"

Clearly the author has not heard of quorum sensing. The notion that bacteria are hyperindividualistic robinson crusoes adrift in their environment is generally not believed anymore.
http://en.wikipedia.org/wiki/Quorum_Sensing

Then the author redefines complexity to be a 'large genome' thing. Of course there really isn't anything stopping bacteria from having ploidy. Deincococcus Radiotolerans, for example, has 4-10 copies of its genome inside itself (for reasons that you might be able to guess from its name).
http://en.wikipedia.org/wiki/Deinococcus_Radiotolerans

And the whole complexity thing is bunk anyways. The actinomyces are such complex bacteria that for many many years scientists thought they were fungi.
http://en.wikipedia.org/wiki/Actinomyces

It's also worth noting, that there certainly are eukarya that are plenty 'complex' that simply don't have mitochondria.
http://en.wikipedia.org/wiki/Giardia

Better explanation: Eukarya were enabled because the Great Oxygen Catastrophe suddenly made being a paired oxidizer (mitochondrion)-reducer (host cell) beneficial. Prior to the oxygen catastrophe, mitochondrial cells would have only been able to survive in homeostasis with a cell that was happy to have its poop (oxygen) taken care of locally... Not to mention that eukarya are generally considered to have branched off of archae... The specialness of eukarya is that they were a unification of two highly divergent branches of the phylogenetic tree.