Aaron Smith, 23, was last seen leaving the hospital on Longfleet Road at 4.55pm on Monday 12 April 2021.
He is described as white, five feet eight inches tall and was wearing a grey tracksuit.
Chief Inspector Matt Baxter, of Dorset Police, said: “We are very worried for Aaron and it is really important that we find him to ensure he is safe and well.
“Officers are carrying out searches of the area in a bid to locate Aaron and I am asking the public to please contact Dorset Police if they have seen him or a man matching the description given.
“Finally, I would like to appeal directly to Aaron if you see this – please call us and let us know where you are as we just want to make sure you are OK.”
Anyone with information or knowledge as to Aaron’s whereabouts is asked to contact Dorset Police at www.dorset.police.uk, via email firstname.lastname@example.org or by calling 101, quoting incident number 12:637.