Oooh, a topic I happen to really love. I find that certain modern anime character designs to be extremely aesthetic, particularly due to better computerized tools for Animation that allow for more complex styles. The theme you'll note in my answers is that I prefer as much realism as 2D animation will allow without hitting that uncanny valley.
(3) There are a lot of characters with heterocrhomia that I find interesting. While I suspect many other AP respondents will mention the Lelouch Lamperouge, or the Kurumi Tokisaki, I am going to mention my favorite heterochromia girl, even if she might be using contacts to do so,
Rikka Takanashi from
Chuunibyou demo Koi ga Shitai! (
Love, Chunibyo & Other Delusions!). That's mainly because it's part of who she is or has chosen to be.
(2)
Violet from
Violet Evergarden. Nothing eccentric, but really well done eyes, -- and another Kyo-Ani production. Here the artists went out of their way to give the main heroine very expressive eyes, and their diligence certainly paid off. Just beautiful.
*
drumroll* (1)
Rikka Takarada from
SSSS.Gridman. (Hmmm, another Rikka.) Yes, we know, she has other assets that are very attractive, certainly, but Rikka has
awesome eyes, such that I immediately fell in love with her for that alone. That deep blue shading with magenta highlights is very pleasing and distinct. Her character design in that regard is superb, and the animators have to be commended for persisting in going with such a difficult to animate feature of gradients and psuedo-gradients, without going overboard. Just look at those
eyes!
*Chef's kiss*
Honorable mention goes to
Major Motoko Kusanagi from
Ghost in the Shell 2045. She would rank higher were this not full CGI, but worth mentioning because the construction of her eye design reminded me me of Rikka above.
Another honorable mention goes to the character
Lacie from the anime of
Beatless. The design for an android's eyes is pretty good, but unfortunately the animated results do not live up to the character designs by Redjuice in the light novels' illustrations, which are excellent.