What's the proper index for querying structures in arrays in Postgres jsonb?

First of all, you cannot access JSON array values like that. For a given json value:

[{"event_slug":"test_1","start_time":"2014-10-08","end_time":"2014-10-12"},
 {"event_slug":"test_2","start_time":"2013-06-24","end_time":"2013-07-02"},
 {"event_slug":"test_3","start_time":"2014-03-26","end_time":"2014-03-30"}]

A valid test against the first array element would be:

WHERE e->0->>'event_slug' = 'test_1'

But you probably don't want to limit your search to the first element of the array. With the jsonb data type in Postgres 9.4 you have additional operators and index support. To index elements of an array you need a GIN index.

The built-in operator classes for GIN indexes do not support "greater than" or "less than" operators > >= < <=. This is true for jsonb as well, where you can choose between two operator classes. The manual:

Name             Indexed Data Type  Indexable Operators
...
jsonb_ops        jsonb              ? ?& ?| @>
jsonb_path_ops   jsonb              @>

(jsonb_ops being the default.) You can cover the equality test, but neither of those operators covers your requirement for >= comparison. You would need a btree index.

Basic solution

To support the equality check with an index:

CREATE INDEX locations_events_gin_idx ON locations
USING gin (events jsonb_path_ops);

SELECT * FROM locations WHERE events @> '[{"event_slug":"test_1"}]';

This might be good enough if the filter is selective enough.
Assuming end_time >= start_time, so we don't need two checks. Checking only end_time is cheaper and equivalent:

SELECT l.*
FROM   locations l
     , jsonb_array_elements(l.events) e
WHERE  l.events @> '[{"event_slug":"test_1"}]'
AND   (e->>'end_time')::timestamp >= '2014-10-30 14:04:06 -0400'::timestamptz;

Utilizing an implicit JOIN LATERAL. Details (last chapter):

• PostgreSQL unnest() with element number

Careful with the different data types! What you have in the JSON value looks like timestamp [without time zone], while your predicates use timestamp with time zone literals. The timestamp value is interpreted according to the current time zone setting, while the given timestamptz literals must be cast to timestamptz explicitly or the time zone would be ignored! Above query should work as desired. Detailed explanation:

• Ignoring time zones altogether in Rails and PostgreSQL

More explanation for jsonb_array_elements():

• PostgreSQL joining using JSONB

Advanced solution

If the above is not good enough, I would consider a MATERIALIZED VIEW that stores relevant attributes in normalized form. This allows plain btree indexes.

The code assumes that your JSON values have a consistent format as displayed in the question.

Setup:

CREATE TYPE event_type AS (
 , event_slug  text
 , start_time  timestamp
 , end_time    timestamp
);

CREATE MATERIALIZED VIEW loc_event AS
SELECT l.location_id, e.event_slug, e.end_time  -- start_time not needed
FROM   locations l, jsonb_populate_recordset(null::event_type, l.events) e;

Related answer for jsonb_populate_recordset():

• How to convert PostgreSQL 9.4's jsonb type to float

CREATE INDEX loc_event_idx ON loc_event (event_slug, end_time, location_id);

Also including location_id to allow index-only scans. (See manual page and Postgres Wiki.)

Query:

SELECT *
FROM   loc_event
WHERE  event_slug = 'test_1'
AND    end_time  >= '2014-10-30 14:04:06 -0400'::timestamptz;

Or, if you need full rows from the underlying locations table:

SELECT l.*
FROM  (
   SELECT DISTINCT location_id
   FROM   loc_event
   WHERE  event_slug = 'test_1'
   AND    end_time  >= '2014-10-30 14:04:06 -0400'::timestamptz
   ) le
JOIN locations l USING (location_id);