Indeed, your oponent clearly does not intend to be kind and nice to others, even when through his loss he is being taught a valuable lesson as to exactly why he lost that certain chess game. However, note that this kind of continuous complaining is rare on chess.com and you will most likely never experience such an opponent ever again while on the duration of this website.
Poor sportsmanship in live chess
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After I first moved to Thailand. I met a guy who was really good at Thai. SO much so, that I commented to a friend of mine that I thought the guy was fluent.
My friend pointed out that it only seemed that way because my Thai was not nearly as good yet.
I think that is what is happening here. They see you make a move that they themselves were unable to imagine, so the only answer, in their mind, is that you must be cheating.
I don't think someone would accuse you of cheating unless, in their mind, they believed it was true. Regardless of if it is true or not.
Clearly taking notes from Mamedyarov.
I remember several years ago I beat someone in live chess who was rated over 300 points higher than I was because he overlooked a simple tactic that won his Queen. He couldn't handle that I saw something he didn't and he claimed the only explanation was that I was cheating. It's pretty funny how stupid people can be sometimes.
It's not a new trend, on the contrary, it's been the one reliable constant of Internet chess since ICS went all commercial and the general public got involved.
I've been accused of cheating as well, because I know opening principles and win in a dozen moves or less against people who break them. I've been accused of cheating because I remember the book lines of openings I've played since high school. I've been accused of cheating because I know certain endgames pretty well and my opponents clearly did not. Ive been accused of cheating for capturing en passant. I've been accused of cheating because my opponents let their time run out.
Honestly, I don't think cheating is as big a problem as people who make accusations of cheating.
It's reprehensible and inexcusable. Nevertheless, it always makes me feel good in some sense to be accused of cheating.
"I've been accused of cheating because my opponents let their time run out."
LMAO! 
45 people have been booted from Chess.com in the last 24 hours for cheating.
that seems like a lot to me.
I still don't understand either the lure of cheating or the fear of it. I played against one who was banned for cheating, and the games were very tough. I had to play really well, and I was still at a disadvantage, and I learned a lot. Meanwhile, the cheaters' ratings will always rise, so they're soon out of your ratings range anyway. So what is the real damage? You lose some rating points? Let them cheat. Use their engines to improve your game on their dime the few times you'll ever play them, and move on. Big deal. I just don't get the overreaction, honestly. If it were for money or titles, that'd be different; but it's not ... it's for online chess.
The things that this site accepts versus what is considered bad behavior just boggles my mind (losing on time is considered poor sportsmanship, and can eventually get you banned, etc.).
Nevertheless, in terms of policing its membership, I think it's a far cry better than any other online chess playing site.
Playing under your real name would certainly reduce the cheating and may make this site a little more personal. Has such an idea been tried other than the chess sites that allow engine use like ICCF and FICGS?
In my way of thinking being accused of cheating is a compliment. Just smile and say thank you. They just need to make up an excuse for why they lost.
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I ended up just disabling chat, because there is no button to report abuse.
It would be as simple as, you press a button and the chat log goes off to review for unsportsmanlike conduct. Easy solution. Since chess.com doesn't have time even for that, it is just not reasonable to have in-game chatting.
Is this a new trend? Two live opponents recently, from different countries, have, when their attacking position fell apart, informed me that I am cheating, cursed at me, and quit. If they were children, I could understand tantrums, but I still wouldn't excuse them... Poor sportsmanship is just that.
I rarely have time for 15/10 chess, but I've been aiming to get more time to do it, instead of falling into the "lots of crazy 5/2 games" trap, which leads to sloppy thinking for me.
My 15/10 rating is about 300 points lower than my correspondance rating, and it's pretty accurate for how I do over the board right now... I have moments of insight, but a single mistake in judgement will sink me, and, if I feel I need to hurry, I am prone to making a mistake in calculation.
As a simple guide for those who think that if their opponent in live chess makes a good move, it must be a "computer move", here's the same advice I got back in the early 90s, when playing on text-based chess servers: someone who is cheating by using computer analysis for their games will have a rating that reflects the strength of the computer. I completely agree that this would be cheating, because players should know if they're up against a human, or a computer.
However, rating-wise, whether they are a computer or a human, their rating reflects the difficulty you would have in fighting them. I am also slightly embarassed to point out that modern computers are MUCH better at live chess than my sorry rating.
This seems stunningly obvious, so I think the only real explanation for the tirade of verbal abuse from people who just lost is that they are intentially being poor sports and doing the online equivalent of kicking over the chessboard and spitting on their opponent.
Is this a new trend? I have been playing chess online in one form or another for over 20 years, and to have 2 recent opponents display this kind of poor sportsmanship (a third one did not accuse me of cheating, but did try very hard to curse me out in broken english, which turned out to be really hilarious, but, again, is symptomatic of the same issue.)
I'm tempted to just turn chat off, but these few represent a minority of opponents, and I never mind chatting about a game with my opponent generally. However, I'm finding that it spoils my rare moments of downtime to end with someone who's a poor sport, and I'm only here for the fun and the practice, so chat may be off in my live games in the future, which is really too bad.
Btw, the latest game in question, that inspired me to write this post:
http://www.chess.com/livechess/game?id=760900092
On move 24, my opponent accused me of cheating, and then launched into a tirade about what a waste of time it was to play me, before resigning in what was still a potentially dangerous position for either side. For what it's worth, I made a mistake in move order early on, so I was actually annoyed by my sloppiness, but wasn't going to let a good cross-castled attacking game go to waste without a fight, and with their bishop blocking their own attack on my not-so-awesomely situated king, I had time to build a counterattack. I'm not sure why my starting my own pawn-storm was the trigger point, from their point of view.