In the farewell scene at the end, Amy says
Well, I think this is about the washing up, personally.
What does this mean?
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It's a joke. The Doctor is saying that he wants Amy and Rory to stop travelling with him and get on with their lives before they come to a sticky end like many of his previous companions.
Amy replies "Well, I think this is about the washing up, personally." - as in the real reason is that the Doctor is fed up of Amy and Rory literally leaving dirty dishes (which is colloquially called the "washing up" in England) lying around or stacked in the sink/kitchen - which is usually a point of contention between housemates. Her tone when saying this also implies she is not entirely serious.
I guess an American might more likely say "this is about (not) taking out the trash".
"Washing up" is the British expression meaning 'doing the dishes'. We sometimes also refer to the pile of dirty dishes waiting to be dealt with as "the washing up".
Amy's relationship with the Doctor is unusual - there may be some attraction between them but she firmly loves Rory and she recognises that the Doctor is always 'running away'.
I understand her use of the term here to suggest that the Doctor is leaving Amy and Rory to escape domesticity. They have a married life now, which includes humdrum domestic chores, and the Doctor doesn't want to be part of it.
Amy is referring to herself and Rory as (metaphorically) being a pile of dirty dishes that the Doctor needs to deal with and put away before he's (metaphorically) able to settle down for the evening.
AMY: Why now?
DOCTOR: Because you're still breathing.
AMY: Well, I think this is about the washing up, personally.
DOCTOR: I mean, you're right, there's still heaps of stuff out there to look at. Do you know, there's a planet whose name literally translates as Volatile Circus? Or maybe there's a bigger, scarier adventure waiting for you in there.
The sense of "washing up" as a way of saying "finishing a difficult task" has been around for quite some time:
“Wash up” has occasionally been used in a positive “finish” or “complete” sense (“‘I had an idea,’ he explained.? ‘Just came to me, riding back. I think I know how I can wash it up.’? He would write it now — tonight!” 1929). But “washed up” (or “all washed up”) has pretty consistently meant “worn out, finished, ruined,” whether it was applied to a personal relationship or the career of someone considered, perhaps a bit prematurely, a has-been (“Once he was the most underestimated man in American politics — a washed-up movie star, it was said, who was too old, too simple and too far right to be President,” Newsweek, 1980).