I would recommend not doing that. The shower is for you the place to let your mind wander. If you want to capture that you will stress yourself and your last place where your mind was free will be gone.

Set your phone just outside the shower with a voice recording app on and recording?

Aqua Notes!

https://www.aquanotepad.com/

There are waterproof pencil and paper sets that you can attach to the wall with suction cups. I have a set and use it pretty frequently when I have some sudden shower inspiration that I don’t want to lose.

I mean, you could set your phone outside the shower and record your shower, just speak out your idea and transcribe it after the shower.

I just convince myself that they are not very good ideas. (RIP Mitch Hedburg).

My younger brother has afriends that is a Brazilian math wizard. He does his best work in the shower. He writes his thoughts on the steam in the shower glass and has a water proof camera nearby to take pictures before they vanish.

Grease pen on the shower wall.

If you have glass partitions try a grease pencil or oil based crayon on the glass(There's another name but I'm brain-farting on it. Oil pastels, maybe. Gah, medication that effects your memory sucks.).

You could maybe fab up a thick glass pane with a plastic backer and keep it in the shower. Hanging it might be tricky. I used to love hitting up the weird leftover building supply type places or even flea markets for weird stuff to repurpose.

The white board idea works, but you'd need one, with pen, that is specifically water proof.

Maybe experiment with different pieces of plastic from your local big box or hardware store and a grease pencil.

If there was ever a good use case for a voice assistant, this would be it.

Bath crayons

I set up a mic on a stand right outside the shower curtain, and try to forget about it

I don't use them in the shower, but you can get wetnotes from stores that sell scuba gear. Something like this should be good https://scubapro.johnsonoutdoors.com/us/shop/accessories/wetnotes-underwater-notebook

So many suggestions for putting your phone near the shower but outside of it... phones are waterproof now. I just bring my phone with me in the shower now, and use my notes app to write down anything I don't want to forget. I was already using it for a shower checklist, since I find it pretty hard to function in the shower. Also doubles as a way to consistently clean my phone, which is a nice plus, I likely wouldn't remember to specifically clean it very often otherwise.

If your phone has a charge port plug, make sure to keep that in during the shower, if it doesn't have a plug, keep the port oriented down, or you may have to wait a couple hours to be able to charge after. I've had my last couple phones in the shower hundreds of times now over the past 10 years, only had it happen twice where the charge port got too wet and had to dry before I could charge it, and neither time was when the battery was low enough for it to matter.

I just make a mental note, like a one word trigger that will remind me of the idea later

Nothing special.

I have a decent quality paper notebook and a pencil attached to it sitting on a shelf nearby the shower (using a softer pencil than your usual HB/2B, say a 4B or 6B) and I can easily reach out for them whenever I need to jot something down. Not worrying at all about messing the paper with water as its sole purpose is to hold the notes for as long as it takes me to finish showering: it being decent/good quality paper, Clairefontaine in my case, it won't instantly break apart when faced with water. I simply don't write under the shower.

I would try an LCD doodle board, I have one I bought at lidl for 3,50€, I doubt it is waterproof but it would be very easy to waterproof it with a vacuum sealer.

You can get bath crayons for kids that let you write directly on the wall of the shower or w/e. Searching brings up lots of options...

I'd just shout out those random ideas and hopefully my partner is right outside the bathroom to hear them.

Rite in rain + pencil

Maybe you can voice record yourself showering and then start talking about it out loud

When I'm in the shower, I write stand up routines I'll never remember or have a reason to use.

Haven't investigated it much, but I've been wondering if you could set your phone or something to transcribe audio and just talk out loud.