MySQL 8.0

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CREATE TABLE t(id INT PRIMARY KEY, myColumn VARCHAR(100));

CREATE UNIQUE INDEX myIndex ON t((CASE WHEN myColumn <> 'myText' THEN myColumn END))

Records: 0  Duplicates: 0  Warnings: 0

INSERT INTO t(id, myColumn)
VALUES(1, 'myText'), (2, 'myText')

Records: 2  Duplicates: 0  Warnings: 0

INSERT INTO t(id, myColumn)
VALUES(3, 'aaaaa')
INSERT INTO t(id, myColumn)
VALUES(4, 'aaaaa')
-- Duplicate entry 'aaaaa' for key 'myIndex'

Duplicate entry 'aaaaa' for key 't.myIndex'

SELECT * FROM t

id
myColumn

1
myText

2
myText

3
aaaaa