You are wise to note that development can be a central issue in relation to the behaviors your daughter is displaying. While it is certainly possible for a four-year-old to display ADHD, care must be taken to have the child undergo a systematic evaluation. The challenge of diagnosing ADHD in a pre-school-age child is that the several clusters (inattentive, hyperactive, impulsive) of symptoms associated with ADHD can be typical of normal pre-schoolers. So, the main issue is not so much whether a pre-schooler happens to be somehwat inattentive, overactive, or prone to impulsive behavior, but rather the degree to which those types of behaviors are occurring and whether the behaviors are an impediment to mastering the normal developmental challenges of the age. An evaluation is very sensible.
Yes, I would have your child evaluated by a child psychologist. By the way, children suffering from anxiety often display ADHD and/or ADD behaviours in social settings as school but when they are in a comfortable and perceived "safe" environment, these behaviours are non-existent. In either case, both these behaviours will require further treatment and the earlier the diagnosis, the better the prognosis.