I recently bought a new dress is the proper way to say it.

They are both acceptable. When talking about the recent past, especially with words like "just", you can consider the event as a finished event (past simple) or as an event at some unspecified recent time before now (present perfect). The choice of the past simple often indicates a more informal tone and / or greater emphasis or excitement. e.g. I have just bought a new coat and I think I will wear it later today. e.g. I just bought this fantastic new coat! What do you think of it?