Is there a word for the ideas you get while showering?
Doubtful there's an English word for that, but I'd be open to words from other languages as well.
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try eureka moment or eureka effect - the original Eureka ("I've found it") moment is named after a myth regarding Archimedes who came upon an epiphany while being in a bath (is that close enough to your shower requirement?). Anyway, Urban Dictionary has the word showertime epiphany for exactly what you're looking for:
An epiphany, a shocking revelation that occurs when doing the most mundane of tasks, such as showering. This could be the solution to a long-unresolved problem.
The word breakthrough is defined as
a sudden increase in knowledge, understanding, etc. : an important discovery that happens after trying for a long time to understand or explain something
Of course, breakthroughs occur in other circumstances besides while showering, but it might be helpful to understand the conditions that are present that often lead to breakthroughs:
The article "Why thinking in the shower may be an ideal model for 'creative pause'” discusses the term creative pause, define by Professor Lajos Székely, in his paper for International Journal of Psychoanalysis, as
the time interval which begins when the thinker interrupts conscious preoccupation with an unsolved problem, and ends when the solution to the problem unexpectedly appears in consciousness.
In that article, it summarizes several factors present while showering that promote creativity while encouraging a temporary reduction in one's attention to the problems the problems at hand.
It goes on to enumerate several factors that might encourage this process:
I'll leave it to the article to provide the details, which can be obtained at the link. (The topic can also be researched through the other references provided within this response.)
The subject can also be found having been discussed by Dr. Edward de Bono, who originated the expression lateral thinking. In his book "Serious Creativity: Using the Power of Lateral Thinking to Create New Ideas", he says
Some of the best results come when people stop to think about things that no one else has stopped to think about.
Obviously creative pause or creative disengagement doesn't only happen in a shower. It describes the conditions that lead to creative ideas; conditions where one can distance oneself from their challenges and cleanse their mind for a moment or longer and refresh their thoughts on the matters that concern them.
Well, there's "Johnny Walker wisdom" (thanks to poet/musician Leonard Cohen). There's a French expression for "brilliance on the staircase" (you know, when you think of the perfect riposte only while leaving the party, and you're on the steps leading down to your car!). Then there are people who love to sing (or think they can sing) in the shower. So why not the franglais douche ideation (or ablution ideation, or showertime synapse firing)?
A rumination - "the act of pondering or meditating" http://www.thefreedictionary.com/rumination