Sorry, but a .38 would be out of the question for 4-legged critters. As for 45 - it depends on which one. A .45 long colt loaded very hotly can suffice, but a .45acp would not.
If you want a semi-auto, then the only one I would consider would be a 10mm with loads at the high end of the range for that caliber (check Buffalo Bore and see if they have any). There are more powerful semi's, but I would not want to lug a Desert Eagle around.
A more suitable handgun would be a revolver. A .357 mag might be ok, but if I were needing defense from blacks or cougars, then I would want something that started with a four. A .44 mag would suffice, as would a .45 long colt. Even a .450 revo from S&W would do fine, but it would be harder to find ammo, and might also be a more expensive to shoot. You can find used .44mags all day long on Gunbroker.com, or over at the Colt and S&W forums. I think a 4" S&W 29 or 629 would fit the bill.....