Depression can manifest itself in many ways. Even to the point, the individual may not even realise they are suffering from depression. It should not be treated lightly. With often multiple layers of issues gathered. Professional advice should be sought.
While hair loss can be an emotive issue. Treating or not is a personal choice. With no definitive right or wrong. Being at peace with your decision is important. That relies on being honest regarding your own motivations for wanting to deal with hair loss. The same applies if deciding to do nothing.