Q:

Do composite key indices improve performance of or clauses

I have a table in MySQL with two columns
id int(11) unsigned NOT NULL AUTO_INCREMENT,
B varchar(191) CHARACTER SET utf8mb4 DEFAULT NULL,

The id being the PK.
I need to do a lookup in a query using either one of these. id in (:idList) or B in (:bList) 
Would this query perform better if, there is a composite index with these two columns in them? 

A:

No, it will not.
Indexes can be used to look up values from the leftmost columns in an index:

MySQL can use multiple-column indexes for queries that test all the columns in the index, or queries that test just the first column, the first two columns, the first three columns, and so on. If you specify the columns in the right order in the index definition, a single composite index can speed up several kinds of queries on the same table. 

So, if you have a composite index on id, B fields (in this order), then the index can be used to look up values based on their id, or a combination of id and B values. But cannot be used to look up values based on B only. However, in case of an or condition that's what you need to do: look up values based on B only.
If both fields in the or condition are leftmost fields in an index, then MySQL attempts to do an index merge optimisation, so you may actually be better off having separate indexes for these two fields.
Note: if you use innodb table engine, then there is no point in adding the primary key to any multi column index because innodb silently adds the PK to every index.

