I know that most of you will say just leave it as it is. However, I would like to know if there is some way to lighten the dark tone. This subject was covered in a 2009 question, but no really helpful information was given. At least none that was helpful to me. Please respond if you have a suggestion other than leave it as it is.
There's the boiling water, aluminum foil, and baking soda method that might lighten it if used shortly.
You probably could brighten it by using lemon juice in a careful manner, with quick 5 minute dips and then carefully examining the results. I'd probably try acetone first...sometimes acetone will brighten a coin, and it's safer. The question is not whether you can brighten it....you can. The question is what that process will reveal. You could end up with some unpleasant results if the removal reveals some old harsh cleaning underneath, or even a few dark spots from old mineral deposits. Or you could get lucky and have a totally pristine surface. It's up to you.
I love cleaning ancient silver and usually default to the aluminum foil and electrolyte method though that would remove all the sulfide patina. Ammonia or citric acid solution swabbed with a q-tip would brighten the high points and help preserve the contrast with the patinated recesses. You could also carefully buff it with a lens cleaning cloth. I'd leave it alone though as it looks quite nice, and that's coming from a habitual coin-cleaner.
My question might be why you don't just sell this coin to someone who likes the natural look and buy a bright one that looks good to you. This is a common type. There are thousands of them that have already been stripped of the tone. There are also people who prefer the look you show here. Certainly you might be able to accomplish what you wish but, unless you are a 'habitual coin cleaner' (meaning you have done this before and have learned when to stop or what a coin that will benefit from cleaning looks like) you stand a good chance of being unhappy. It is unlikely that you will 'improve' the coin more than just 'change' it. Also, cleaning it will remove it from the slab for which you paid extra. Many of us remove coins from slabs because we like raw coins better and regret that someone else had them slabbed. Unless the slab has a comment on it that would make it hard to sell, I'd sell it to someone who wants a toned and slabbed Alexander.
You are right. Why don't you just leave it as is? Your coin has some beautiful toning. Nice score btw
The OP coin has beautiful even dark toning. If the toning is somehow removed, it will no longer be "even" and it is highly unlikely to remain "beautiful." @dougsmit is right. If you don't love it, there are many out there who would. Put it on the CT for-sale page: https://www.cointalk.com/forums/for-sale/ and sell it and use the proceeds to buy a coin you prefer.