Joining arrays within group by clause

UNION ALL

You could "counter-pivot" with UNION ALL first:

SELECT name, array_agg(c) AS c_arr
FROM  (
   SELECT name, id, 1 AS rnk, col1 AS c FROM tbl
   UNION ALL
   SELECT name, id, 2, col2 FROM tbl
   ORDER  BY name, id, rnk
   ) sub
GROUP  BY 1;

Adapted to produce the order of values you later requested. The manual:

The aggregate functions array_agg, json_agg, string_agg, and xmlagg, as well as similar user-defined aggregate functions, produce meaningfully different result values depending on the order of the input values. This ordering is unspecified by default, but can be controlled by writing an ORDER BY clause within the aggregate call, as shown in Section 4.2.7. Alternatively, supplying the input values from a sorted subquery will usually work.

Bold emphasis mine.

LATERAL subquery with VALUES expression

LATERAL requires Postgres 9.3 or later.

SELECT t.name, array_agg(c) AS c_arr
FROM  (SELECT * FROM tbl ORDER BY name, id) t
CROSS  JOIN LATERAL (VALUES (t.col1), (t.col2)) v(c)
GROUP  BY 1;

Same result. Only needs a single pass over the table.

Custom aggregate function

Or you could create a custom aggregate function like discussed in these related answers:

• Selecting data into a Postgres array
• Is there something like a zip() function in PostgreSQL that combines two arrays?

CREATE AGGREGATE array_agg_mult (anyarray)  (
    SFUNC     = array_cat
  , STYPE     = anyarray
  , INITCOND  = '{}'
);

Then you can:

SELECT name, array_agg_mult(ARRAY[col1, col2] ORDER BY id) AS c_arr
FROM   tbl
GROUP  BY 1
ORDER  BY 1;

Or, typically faster, while not standard SQL:

SELECT name, array_agg_mult(ARRAY[col1, col2]) AS c_arr
FROM  (SELECT * FROM tbl ORDER BY name, id) t
GROUP  BY 1;

The added ORDER BY id (which can be appended to such aggregate functions) guarantees your desired result:

a | {1,2,3,4}
b | {5,6,7,8}

Or you might be interested in this alternative:

SELECT name, array_agg_mult(ARRAY[ARRAY[col1, col2]] ORDER BY id) AS c_arr
FROM   tbl
GROUP  BY 1
ORDER  BY 1;

Which produces 2-dimensional arrays:

a | {{1,2},{3,4}}
b | {{5,6},{7,8}}

The last one can be replaced (and should be, as it's faster!) with the built-in array_agg() in Postgres 9.5 or later - with its added capability of aggregating arrays:

SELECT name, array_agg(ARRAY[col1, col2] ORDER BY id) AS c_arr
FROM   tbl
GROUP  BY 1
ORDER  BY 1;

Same result. The manual:

input arrays concatenated into array of one higher dimension (inputs must all have same dimensionality, and cannot be empty or null)

So not exactly the same as our custom aggregate function array_agg_mult();