My brutal assesment is that this is the typical watch a dealer will produce. Over polished and incorrectly appointed.
Now it is not all bad news. £4000 for a 105.003 is not extortionate in these markets. Sure, if you hunt and scrounge you might find one for £3000 but it's going to have a bucket of problems.
This ones problems are easy to make, if not good, then pretty good.
All the bezel discussion is a bit of a distraction. It's just money, now that market prices have risen to £600-1400
And drawn them out of people's parts drawers! What I mean is, it's up to you where you go, all options are available. Six months ago you couldn't find them.
The case is incorrectly polished, as well as over polished. A good workshop will re finish the sides to correct spec and genreraly make the best of what you have got. It will be a far better result than what you have now.
The pushers are so completely wrong they ruin the look of the watch for me. Again, service items easily replaced.
Crown can be sorted by Micheal e as suggested.
The hands are modern service replacements. Again a good workshop will age them. They won't fool some, but they will make the watch much easier to live with and much more attractive.
In these times, finding a 105.003 is very difficult. Attractive ones sell for a lot of money, and this brings onto the market poor examples chasing those high prices.
This watch could have been a lot worse at this money. It looks an original case/movement/dial combination, and I can't tell you how rare that is becoming. Clearly it has been worked on by some one without specialised speedmaster knowledge and so I would send it straight to STS for a service and light case refinish, and hand colour adjustment.
Don't beat yourself up.
This can be a decent watch for not much further outlay.
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