The Pillowed Mastermorphix (sometimes called Master Pyramorphinx) is a monster. You can get two versions, single-colour or 4-colour as you can see in the image.
This puzzle changes shape when you mix it up. For the single-colour version, you just have to return it to its original shape - not as easy as it sounds. The four-colour version needs a lot more work.
Despite everything your hands and eyes tell you, the Pillowed Mastermophix is essentially a 3x3x3 cube with odd-shaped pieces. Once you get your head around this, solving gets a whole lot easier. The 4 tips of the tetrahedron are corners as are the ickle triangles in the centre of each 'face'. Try to work out how to map each piece shape to its 3x3x3 cube equivalent and you will get there.
One major difference from the cube is that it is possible to have twisted centre pieces and you can get into a position where everything but a singe centre-piece is solved. This is a frustrating puzzle but is so well made and so satisfying to complete that I recommend it to anybody that wants to move on from the cube.