How can I create a unique index in Oracle but ignore nulls?

I don't know how many people still get directed to this answer, but at least in the latest version of oracle, you're allowed to have multiple rows with null on a unique index and the accepted answer isn't necessary

We can do this with a function-based index. The following makes use of NVL2() which, as you know, returns one value if the expression is not null and a different value if it is null. You could use CASE() instead.

SQL> create table blah (name varchar2(10), email varchar2(20))
  2  /

Table created.

SQL> create unique index blah_uidx on blah
  2      (nvl2(email, name, null), nvl2(name, email, null))
  3  /

Index created.

SQL> insert into blah values ('APC', null)
  2  /

1 row created.

SQL> insert into blah values ('APC', null)
  2  /

1 row created.

SQL> insert into blah values (null, '[email protected]')
  2  /

1 row created.

SQL> insert into blah values (null, '[email protected]')
  2  /

1 row created.

SQL> insert into blah values ('APC', '[email protected]')
  2  /

1 row created.

SQL> insert into blah values ('APC', '[email protected]')
  2  /
insert into blah values ('APC', '[email protected]')
*
ERROR at line 1:
ORA-00001: unique constraint (APC.BLAH_UIDX) violated


SQL>

Edit

Because in your scenario name will always be populated you will only need an index like this:

SQL> create unique index blah_uidx on blah
  2      (nvl2(email, name, null), email)
  3  /

Index created.

SQL>