Here's a tale https://www.revk.uk/2018/07/air-conditioning-at-home-planning.html I suspect if this is pursued the complainer may not ultimately win, and bad-blood will be created. IMHO there are numerous legal ways a disgruntled neighbour can get revenge. Planting 30' bamboo down a boundary for one - totally outside any high hedge legislation as it is a grass and not covered. Live and let live is mostly best. How about a nice blind on that window? it's not as if the view without the aircon is that great!
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