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Comments are the dirtiest part of WordPress. They're on by default, often only styled as an afterthought by theme developers, and universally derided as having poor user experience.
Entire companies have been built around the idea of making comments easier, prettier, more unified across platforms. The LiveFyres and Disquses of the world compete directly with the Facebooks and Jetpacks of the world who try to power prettier comments from their own platforms.
Every single one comes up short.
Owning Your Data
I run WordPress because I want to be in complete control of my content. I want to own the information I publish, and hosting on someone else's platform means not only do they have access to my work, but they can integrate it with their internal analytics and aggregation products. Likewise, owning my own data means no one can arbitrarily take me offline without my involvement.
When you use a 3rd party service to power you comments, you are giving up control over your content - and your visitors' identities - to said third party.[ref]In the interest of full disclosure, I currently use Jetpack Comments on this site. Not because I approve of a third-party comment system, but because it's effectively the lesser evil available.[/ref]
I've seen people proudly use a hosted commenting system for months, only to suddenly find comments on older articles going missing. Or to suddenly have their host lock up data and demand payment for continued use or a data export.
Someone else owning my data scares the crap out of me.
Comments Still Suck
At the same time, the comment features in WordPress core today are absolutely horrible. Comment discussions can thread at most 10 levels deep, meaning there is an inherent limit to any conversation you may attempt to have with readers.
Threaded comments are impossible to manage - if a visitor accidentally comments on the wrong thread, the only way to properly reassign the comment is with direct database manipulation.
Despite the best efforts of theme developers, and even the third party systems, the editing experience of actually drafting a comment is incredibly wanting. Stock comments allow a subset of HTML to be used directly. Some of the nicer comment systems provide access to rudimentary WYSIWYG tools. The difference between drafting a post and drafting a comment in WordPress, however, is stark.
Comments Are Content, Too!
In many contexts, comments are just as - if not moreso - important as the original content itself. More than once, I've even had the total wordcount of comment threads dwarf that of the post that inspired them.
But theme developers still relegate comments to the footer of a post.
One of the cooler features of Medium (that I often hear requested for WordPress) is the ability to comment on individual statements in a post. As each article often contains more than one thought worthy of comment, this would be incredible.
Comments, on the other hand, might also contain more than one statement worthy of comment and likewise should allow per-statement comment threads.
My point - comments are just as important as their parent article. They need to be elevated to such a position, both within the theme and within WordPress itself.
A Way Forward
Ultimately, I would love to see a WordPress release themed around collaboration.
Collaborative content creation, editing, publication, revision.
Post-publication comments are also a form of collaboration, and they are in drastic need of some refactoring and re-imagining.
bbPress, the WordPress bulletin board plugin, allows individual threaded conversations. I would love to see posts and comments reworked (either in core or in a plugin) into a threaded conversation - treating comments on par with the post itself rather than as after-the-fact asides.
Likewise, a WordPress theme that properly elevates comments from the post footer to real content status would be a great way to lead the charge of creating a rich, collaborative experience.
What would you do to improve WordPress comments?