As I've said before, the 5e Beastmaster ranger looks mehsville, baby. So instead of being an archetype for the ranger, what if it was offered as a feat anybody could take if they wanted an animal buddy?
It might look something like this:
Prerequisite: Proficiency with the Animal Handling skill.
You gain an animal companion that accompanies you on your adventures and is trained to fight alongside you:
- Choose a beast that is no larger than Medium and that has a challenge rating of 1/2 or lower. Add your proficiency bonus to the beast’s AC, attack rolls, and damage rolls, as well as to any saving throws and skills it is proficient in. It is proficient in any saving throws you are proficient in. Its hit point maximum equals its normal maximum or four times your level, whichever is higher.
- The beast obeys your commands as best as it can. It takes its turn on your initiative, though it doesn’t take an action unless you command it to. On your turn, you can verbally command the beast where to move (no action required by you). You can use a bonus action to attempt to verbally command it to take the Attack, Dash, Disengage, Dodge, or Help action. This attempt is a Wisdom (Animal Handling) check made against a DC of 15.
- When you cast a spell targeting yourself, you can also affect your beast companion with the spell if the beast is within 30 feet o f you.
- If the beast dies, you can return it to life by spending 3 hours conducting a special resurrection ritual. Alternately, you can obtain another one by spending 8 hours magically bonding with another beast that isn’t hostile to you, either the same type of beast as before or a different one.
Okay, so that's one way to do it. But honestly? If someone really wanted an animal companion I'd probably just use the rules (well, guidelines really) for allies in the DMG and just have the beast level up when the PCs do.