The flash built-ins don't have serialization methods. (there are some major complexity issues - e.g. reconstruction of a partially serialized display list)
It is possible instead to write the subset of functionality that you want to serialize - for BitmapData, a straightforward path is to transfer to Vector and then to Array, then serialize the Array. If only Flash needs to use the data you can also use the ByteArray methods that take or emit Objects and get a speed boost over haXe native, but again, not all API is directly supported.