Can I use sandoz tobramycin for my 8 month old baby? Are there any serious side effects to look out for?

No. You should not be using anything on your baby without medical supervision.
Unable to assess. Has the baby seen the doctor? Other healthcare professionals may disagree, but even if the baby is old enough to safely receive the drug itself, it’s impossible to answer your question without a description of symptoms and a physical examination. The reason: the drug may be inappropriate, in which case the baby will be at risk of side effects without benefit and deprived of effect.
For what ? If the kids eyes are a little crusty it could be just allergies, a virus, irritants from the air or a real infection. Using an expensive eyedrop for the wrong reason is a waste of good medicine & can delay proper treatment. A relatively cheep simple eye drop may be all that is needed. At least call the prescribing doc & see what they think about the condition before use.
I would say no- -there are very serious side effects such as loss of hearing. Maybe you should get a second would be one of my last choices for antibiotics, especially if no others have been tried. You should be seeing a pediatrician of repute, & ask why this was the choice that was made.