All modern turbo engines should use fully synthetic oils due to the extreme temperatures generated by the turbo. Mineral oils are much more susceptible to 'coking' where the heat causes the oil to crystalise if it sits in the turbo for too long (this happens if you don't give your engine a cooling down period after using the turbo and before turning it off).
Any grade between 0w30 and 5w40 is typical for modern turbocharged engines. For reference, the numbers refer to the cold and hot viscosity levels of the oil. A higher viscosity (represented by a higher number) provides better lubrication, but the downside is it takes longer to circulate through the engine - like treacle. Modern synth oils are engineered to have different viscosity levels at different temperatures, the first number before the W is how viscous it is at cold temps - usually very low so that from start-up the oil can flow through the engine like water and start protecting the metal on metal components. The second number is the viscosity at normal operating temperature - once distributed through the engine the oil becomes more viscous and coats the components better to aid lubrication.
All engines have slightly different performance characteristics, so I would stick with BMWs recommended - especially under warranty. Audi notoriously denied warranty claims to customers who did not use the Audi recommended grades of oil in the 1.8T and who ended up with oil-sump blockages due to coking (even though it is widely accepted it was due to bad design of the oil-pickup pipe and filter).