RAW: Roll the damage dice twice
The standard rule for critical hits from the basic rules states:
When you score a critical hit, you get to roll extra dice for the attack's damage against the target. Roll all of the attack's damage dice twice and add them together. Then add any relevant modifiers as normal. To speed up play, you can roll all the damage dice at once.
This is the 'official' version of the critical hit rule. When you score a critical hit, you simply roll twice as many dice as you would otherwise then adds your standard modifiers to the result.
Other variations
There are a lot of variations on the standard rule, have a look through our critical-hit questions for just a few of them. I'll try to list a few of the common ones I have encountered and their advantages.
Double the number rolled
This is the system used by your DM, it is a common variation on the standard rule. See What are the statistical implications of doubling damage on crit instead of doubling the dice rolled? for a full breakdown on the mathematical differences.
- Advantages:
- Reduces the total number of dice rolled
- Same average as the standard rule.
- Neutral:
- Requires players to do multiplication instead of addition
- Disadvantages:
- Increases the variation (swinginess) of the roll
- Low damage rolls are anticlimatic
Maximum + Rolled
This is the system you suggest and is the one I personally prefer as it feels the most satisfying.
- Advantages:
- Reduces the total number of dice rolled
- Critical hits guarantee additional damage
- Neutral:
- Increases the overall damage output (combat's are shorter)
- Disadvantages:
- Calculating maximum damage on a complex dice pool can be slow
- Typically advantages monsters as they make more attacks
Notably, any system that increases the average damage of critical hits by more than the official method can affect game balance. It tends to increase the power of martial classes relative to casters who don't make attack rolls, but force targets to make saving throws. Similarly, the increase to damage advantages monsters as damage dealt to PCs is more meaningful than damage dealt to monsters.
Lingering Injuries
There is an option in DMG for Lingering Injuries. These trigger on a critical hit and can apply ongoing wounds or effects.
- Advantages:
- Add be added to any of the other variant rules
- Makes critical hits always feel meaningful
- Disadvantages:
- An extra thing to roll for slows down the game
- Wounds rarely effect enemies during the fight
- Ongoing wounds quickly add up in long campaigns
Other Bonus effects
I've seen a variety of systems from DMs that apply various additional effects on critical hits. Some examples are:
- "Called shot" effect; disarm, cut off a limb, cripple or blind the target.
- Target must make a constitution save vs damage received or be stunned until the end of its next turn.
- Advantage on the next hit against the target.
- Trigger "massive damage" instant death rule
- Any other status effect you can think of
Each rule has its own advantages and disadvantages and I'm not recommending them other than letting you know that they have been done. The point is that there are unlimited ways to play critical hits and you are welcome to come up with your own.
How to choose a rule
Which rule you are going to use is something that should be discussed in a session 0. Agree on a rule as a group and implement it consistently. If issues arise you can revisit the conversation and agree to change (or not) going forward. So long as everyone is on the same page and having fun then you are playing the correct rule.
Thanks for Medix2 and Louis Wasserman in the comments for their excellent advice.