Yes, there are places in Bali that are similar to TwinPalms in Ohuket, but it depends on what you want, like and budget.
You can rent lovely villas (like those in Phuket, but much cheaper than Phuket ) and then there are lovely hotels.
If it's lush foliage you want and jungle views, then head for Ubud. Some nice hotels there are Royal Pita Maha, Ibah, Ubud Hanging Gardens, etc...
If you want good food, upmarket shopping, then head to Seminyak - and you're not far from the beach. There are mainly villas in Seminyak though. Some nice villas are:
Villa Kubu, The Ahimsa, The Amasya, etc...
If not, some hotels there are:
The Legian, The Oberoi, Sofitel Seminyak The former two have their own beach, and I think Sofitel as well, but I'm not sure about that.
Alternatively, if you want the Laguna type area of Phuket -- but wider and nicer, then head for Nusa Dua. Pristine lawns, landscaped pools, big name hotel chains. I have stayed at Nusa Dua while it's very nice, I find it too "artificial" for my personal liking.
You'll be able to find nice stylish, relaxed environments In all these 3 areas...Hope this helps.
For us the Kuta Paradiso has been fantastic over many years. It's really close to Kuta markets/square and Matahari and new shopping centre etc. It is a 4 star hotel with great pools and great ambience. Also, it's not as big as the others, more of a boutique style and it doesn't front onto a beach which is a plus!
We stayed at The Courtyard Hotel & Apartments in Legian last Sep-Oct and loved it. It's only a small boutique hotel with 18 rooms, which range from basic rooms to 1 and 2 bedroom apartments. The 2 bedroom apartments looked gorgeous but we were just as happy with our basic rooms (our party occupied 5 all up which was heaps of fun). The look is very contempory and stylish, the food was delicious and the guys working there were terrific! Even though it's on a pretty busy road, it's set back and you seriously hardly hear a thing. It's very quiet and relaxed! The website is www.courtyard-bali.com but I can email you some of our pics if you'd like. My email address is firstname.lastname@example.org