Well this is just one of my improvisations I recorded, I do improvise almost everyday and for hours.
This is just one piece I could improvise while improvising other things, but I don't record them all the time because I forgot to do it heheeh I always tell myself to do it like, every time I will improvise I must record, but then it takes some time to set it all up, and I never do it hahah
a friend of mine asked me to improvise something on Beethoven's "appassionata", and I had a quick look at the score and took the very beginning, which is just a simple jump from note to note in F minor with little added in-between, then I put it together with my ideas and my feels of that moment, and every now and then used the theme to give to the whole piece a better sense of togetherness.
For the second question/idea, it is a big no, I wouldn't write down an Improvisation, would be pointless and lose the improvisation character itself which consists in the fact that it is made on the spot, it's not something written out, so you had thoughts on it, it is just "free speech", and it has to be that way.
I could improvise you millions different ways on the same theme, in this case of Beethoven, and this would mean I'd need to notate all them? No way! hehe
I dont really care to write down my improvisations, as I said, if i want to make a record of them, I just go and record it. I have my written music and my improvised music. Sometimes while I am improvising it happens that I want to sue that idea for a work, so I take the pen and the paper and write it down and compose the piece while improvising it, that's why it takes me not much time to write something, if I really need it.
But, besides all that I am glad you liked my music! :)