If you feel bohemian whilst in Palma, Son Armadams is definitely the place to go. Hilly narrow streets, its proximity to both the sea and the forest and the views, it all adds up to its bohemian vibe. This is a cute quiet neighbourhood bordering close to the seaside but away from all the hassle and bustle of Passeig Marítim. Writers such as Camilo José Cela lived in the area for some years and it is still a popular area for writers to live in. It lies at the foot of Bellver hill so you could combine this area with your visit to the castle.
It still maintains its XIX and XX century essence and it looks as if time had stopped here, allowing us to enjoy a peaceful walk around beautiful mansions built in the years of its full swing. It used to be, believe it or not, where the bourgeoisie of Palma had their summer residence, as Son Armadams was originally separated from the rest of the city. Most houses are massive and have trees and gardens. I personally find it very relaxing to walk around this area, north of Carrer de Joan Miró.
Unfortunately, both El Terreno and Son Armadams share the story of decadence. They were popular a few centuries back and during the tourism boom of Mallorca in the 60s and 70s many bars, restaurants and clubs opened in the surroundings. In recent years, due to lack of investment by the local authorities and decreased popularity, many of these businesses have closed down and residents do not visit these beautiful neighbourhoods as often as previous generations used to. Some houses are abandoned or being restored, although I personally think this will be a sought-after location for residential property in a few years due to the size of the houses, its proximity to both the forest and the seaside and its quietness.
If you have a chance, visit Son Armadams and let me know what you think of it as it is most certainly not in your travel guides!