Amazon Web Service CloudWatch custom metrics with dimensions

Do note that you can retrieve aggregated value from API, as well as plot a graph in CloudWatch using a math expression. See Using metric math

I should probably also add that you can NOT use metric math in alarms.

update: as @Brooks said Amazon CloudWatch Launches Ability to Add Alarms on Metric Math Expressions

All in all pretty restricted and user-unfriendly compared e.g. to DataDog.

Yes, you can add dimensions such as you described to your custom metrics.

However, CloudWatch is NOT able to aggregate across these dimensions, as it doesn't know the groups of these dimensions. Basically:

Amazon CloudWatch treats each unique combination of dimensions as a separate metric. For example, each call to mon-put-data in the following figure creates a separate metric because each call uses a different set of dimensions. This is true even though all four calls use the same metric name (ServerStats).

See more information about dimensions in CloudWatch here