I know that the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$ is indeed $x^4-10x^2+1$ (as shown multiple times already) because of the fact that it is (1) monic and (2) irreducible in $\mathbb Q$.
Now i am asked to find the minimal polynomials for $\sqrt{2}+\sqrt{3}$ over $\mathbb R$ and $\mathbb C$ aswell.
I used the same technique as for $\mathbb Q$ but halted as soon as i reached the irreducible polynomial, which in this case is $p(x)=x-\sqrt{2}-\sqrt{3}$.
The polynomial $p(x)$ is the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb R$ and $\mathbb C$. It should be trivial for $\mathbb R$ and $\mathbb C$ because $p(x)$ is indeed monic and irreducible by the definition of $\mathbb R$ and $\mathbb C$ (since all polynomials of $deg(1)$ are irreducible).
Is my thought process correct?
You rightly noticed that $\sqrt2+\sqrt3\in\Bbb R$ (aswell as $\sqrt2+\sqrt3\in\Bbb C$). And, indeed, for every $\alpha\in L$, where $L$ is a field, the minimal polynomial of $\alpha$ over $L$ is just $f(x)=x-\alpha\in L[x]$. This is both (1) monic and (2) irreducible as you showed by yourself.
To answer your question directly: yes, your thought process is correct.