Is the word completamente the same in Brazilian and European Portuguese?

Yes

The spelling, meaning, and grammatical function of "completamente" are identical in both Brazilian and Continental Portuguese. The only difference lies in pronunciation.

In Brazilian Portuguese, the vowels are generally more open and clearly articulated, giving the word a more melodic rhythm. In Continental Portuguese, unstressed vowels are heavily reduced or "swallowed" (a process called vowel reduction). Consequently, a Portuguese person might pronounce the word with much shorter, almost silent "e" sounds, making it sound more clipped or "compressed" (e.g., sounding closer to complet'ment').

Brazilian Portuguese Examples

1. O meu celular está completamente sem bateria. (My cell phone is completely out of battery.)
2. Eu fiquei completamente perdido no shopping. (I got completely lost in the mall.)
3. A comida estava completamente fria. (The food was completely cold.)
4. Ele é completamente maluco. (He is completely crazy.)
5. A rua estava completamente deserta. (The street was completely deserted.)

Continental Portuguese Examples

1. O meu telemóvel está completamente sem bateria. (My mobile phone is completely out of battery.)
2. Eu fiquei completamente perdido no centro comercial. (I got completely lost in the shopping center.)
3. A comida estava completamente fria. (The food was completely cold.)
4. Ele é completamente doido. (He is completely crazy.)
5. A rua estava completamente deserta. (The street was completely deserted.)