@amarakon You mean 'TOP LEFT HALF BRACKET' (U+2E22), 'TOP RIGHT HALF BRACKET' (U+2E23), 'BOTTOM LEFT HALF BRACKET' (U+2E24) and 'BOTTOM RIGHT HALF BRACKET' (U+2E25). OK, I'll add them.

Here is what I include in Common Serif compare to other fonts.

Can you make them shorter? If the half bracket is actually half the height of a regular bracket, it looks extremely weird in text.

@amarakon I put a question on different forums about the case. According to Adam Twardoch the half brackets should touch or overlap slighly like in the Brill and Gentium (see the picture above). Can you give me example of use of half brackets?

@amarakon Here is the information which I found on Wikipedia:

Quine corners ⌜⌝ and half brackets ⸤ ⸥ or ⸢ ⸣

The Quine corners ⌜ and ⌝ have at least two uses in mathematical logic: either as quasi-quotation, a generalization of quotation marks, or to denote the Gödel number of the enclosed expression.

Half brackets are used in English to mark added text, such as in translations: "Bill saw ⸤her⸥".

In editions of papyrological texts, half brackets, ⸤ and ⸥ or ⸢ and ⸣, enclose text which is lacking in the papyrus due to damage, but can be restored by virtue of another source, such as an ancient quotation of the text transmitted by the papyrus. For example, Callimachus Iambus 1.2 reads: ἐκ τῶν ὅκου βοῦν κολλύ⸤βου π⸥ιπρήσκουσιν. A hole in the papyrus has obliterated βου π, but these letters are supplied by an ancient commentary on the poem. Second intermittent sources can be between ⸢ and ⸣. Quine corners are sometimes used instead of half brackets.

@amarakon Another position from Unicode forum:

Are you bothered that the top of the bottom half brackets doesn't line up with the bottom of the top half brackets in two of these typefaces? These aren't meant to be drawing tools.
The specifications for this block of characters (https://unicode.org/charts/PDF/U2E00.pdf) says the following interesting things:

Fonts
The shapes of the reference glyphs used in these code charts are not prescriptive. Considerable variation is to be expected in actual fonts.
Half brackets
These form a set of four corner brackets and are used editorially. They are distinguished from mathematical floor and ceiling characters. Occasionally quine corners are substituted for half brackets.
2E22 ⸢ TOP LEFT HALF BRACKET
→ 2308 ⌈ left ceiling
→ 231C ⌜ top left corner
→ 300C 「 left corner bracket
2E23 ⸣ TOP RIGHT HALF BRACKET
2E24 ⸤ BOTTOM LEFT HALF BRACKET
2E25 ⸥ BOTTOM RIGHT HALF BRACKET

Note the alternate characters for other uses.

I put a question on different forums about the case. According to Adam Twardoch the half brackets should touch or overlap slighly like in the Brill and Gentium (see the picture above).

Ultimately, it’s the designer’s (your) choice. But in my opinion, they look a lot better when they don’t touch or overlap. I don’t have an explanation for it, it just looks better that way in my eyes. Perhaps you could make a character variant that users could enable if they wanted to.